ParsedMaterial

coupled_variables

C++ Type:std::vector<VariableName>

Controllable:No

constant_names

C++ Type:std::vector<std::string>

Controllable:No

Description:Vector of constants used in the parsed function (use this for kB etc.)

constant_expressions

C++ Type:std::vector<std::string>

Controllable:No

Description:Vector of values for the constants in constant_names (can be an FParser expression)

material_property_names

C++ Type:std::vector<std::string>

Controllable:No

Description:Vector of material properties used in the parsed function

postprocessor_names

C++ Type:std::vector<PostprocessorName>

Controllable:No

Description:Vector of postprocessor names used in the parsed function

(modules/combined/examples/phase_field-mechanics/Pattern1.i)

#
# Pattern example 1
#
# Phase changes driven by a combination mechanical (elastic) and chemical
# driving forces. In this three phase system a matrix phase, an oversized and
# an undersized precipitate phase compete. The chemical free energy favors a
# phase separation into either precipitate phase. A mix of both precipitate
# emerges to balance lattice expansion and contraction.
#
# This example demonstrates the use of
# * ACMultiInterface
# * SwitchingFunctionConstraintEta and SwitchingFunctionConstraintLagrange
# * DerivativeParsedMaterial
# * ElasticEnergyMaterial
# * DerivativeMultiPhaseMaterial
# * MultiPhaseStressMaterial
# which are the components to se up a phase field model with an arbitrary number
# of phases
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 80
  ny = 80
  nz = 0
  xmin = -20
  xmax = 20
  ymin = -20
  ymax = 20
  zmin = 0
  zmax = 0
  elem_type = QUAD4
[]

[GlobalParams]
  # CahnHilliard needs the third derivatives
  derivative_order = 3
  enable_jit = true
  displacements = 'disp_x disp_y'
[]

# AuxVars to compute the free energy density for outputting
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    additional_free_energy = cross_energy
  [../]
  [./cross_terms]
    type = CrossTermGradientFreeEnergy
    variable = cross_energy
    interfacial_vars = 'eta1 eta2 eta3'
    kappa_names = 'kappa11 kappa12 kappa13
                   kappa21 kappa22 kappa23
                   kappa31 kappa32 kappa33'
  [../]
[]

[Variables]
  # Solute concentration variable
  [./c]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0
      max = 0.8
      seed = 1235
    [../]
  [../]

  # Order parameter for the Matrix
  [./eta1]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  [../]
  # Order parameters for the 2 different inclusion orientations
  [./eta2]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]
  [./eta3]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]

  # Mesh displacement
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]

  # Lagrange-multiplier
  [./lambda]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1.0
  [../]
[]

[Kernels]
  # Set up stress divergence kernels
  [./TensorMechanics]
  [../]

  # Cahn-Hilliard kernels
  [./c_res]
    type = CahnHilliard
    variable = c
    f_name = F
    args = 'eta1 eta2 eta3'
  [../]
  [./time]
    type = TimeDerivative
    variable = c
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 1
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    args = 'eta2 eta3 c'
    mob_name = L1
    f_name = F
  [../]
  [./ACInterface1]
    type = ACMultiInterface
    variable = eta1
    etas = 'eta1 eta2 eta3'
    mob_name = L1
    kappa_names = 'kappa11 kappa12 kappa13'
  [../]
  [./lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name   = h1
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 2
  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    args = 'eta1 eta3 c'
    mob_name = L2
    f_name = F
  [../]
  [./ACInterface2]
    type = ACMultiInterface
    variable = eta2
    etas = 'eta1 eta2 eta3'
    mob_name = L2
    kappa_names = 'kappa21 kappa22 kappa23'
  [../]
  [./lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name   = h2
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 3
  [./deta3dt]
    type = TimeDerivative
    variable = eta3
  [../]
  [./ACBulk3]
    type = AllenCahn
    variable = eta3
    args = 'eta1 eta2 c'
    mob_name = L3
    f_name = F
  [../]
  [./ACInterface3]
    type = ACMultiInterface
    variable = eta3
    etas = 'eta1 eta2 eta3'
    mob_name = L3
    kappa_names = 'kappa31 kappa32 kappa33'
  [../]
  [./lagrange3]
    type = SwitchingFunctionConstraintEta
    variable = eta3
    h_name   = h3
    lambda = lambda
  [../]

  # Lagrange-multiplier constraint kernel for lambda
  [./lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
    epsilon = 1e-6
  [../]
[]

[Materials]
  # declare a few constants, such as mobilities (L,M) and interface gradient prefactors (kappa*)
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'M   kappa_c  L1 L2 L3  kappa11 kappa12 kappa13 kappa21 kappa22 kappa23 kappa31 kappa32 kappa33'
    prop_values = '0.2 0        1  1  1   2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00   '
  [../]

  # We use this to output the level of constraint enforcement
  # ideally it should be 0 everywhere, if the constraint is fully enforced
  [./etasummat]
    type = ParsedMaterial
    property_name = etasum
    coupled_variables = 'eta1 eta2 eta3'
    material_property_names = 'h1 h2 h3'
    expression = 'h1+h2+h3-1'
    outputs = exodus
  [../]

  # This parsed material creates a single property for visualization purposes.
  # It will be 0 for phase 1, -1 for phase 2, and 1 for phase 3
  [./phasemap]
    type = ParsedMaterial
    property_name = phase
    coupled_variables = 'eta2 eta3'
    expression = 'if(eta3>0.5,1,0)-if(eta2>0.5,1,0)'
    outputs = exodus
  [../]

  # matrix phase
  [./elasticity_tensor_1]
    type = ComputeElasticityTensor
    base_name = phase1
    C_ijkl = '3 3'
    fill_method = symmetric_isotropic
  [../]
  [./strain_1]
    type = ComputeSmallStrain
    base_name = phase1
    displacements = 'disp_x disp_y'
  [../]
  [./stress_1]
    type = ComputeLinearElasticStress
    base_name = phase1
  [../]

  # oversized phase
  [./elasticity_tensor_2]
    type = ComputeElasticityTensor
    base_name = phase2
    C_ijkl = '7 7'
    fill_method = symmetric_isotropic
  [../]
  [./strain_2]
    type = ComputeSmallStrain
    base_name = phase2
    displacements = 'disp_x disp_y'
    eigenstrain_names = eigenstrain
  [../]
  [./stress_2]
    type = ComputeLinearElasticStress
    base_name = phase2
  [../]
  [./eigenstrain_2]
    type = ComputeEigenstrain
    base_name = phase2
    eigen_base = '0.02'
    eigenstrain_name = eigenstrain
  [../]

  # undersized phase
  [./elasticity_tensor_3]
    type = ComputeElasticityTensor
    base_name = phase3
    C_ijkl = '7 7'
    fill_method = symmetric_isotropic
  [../]
  [./strain_3]
    type = ComputeSmallStrain
    base_name = phase3
    displacements = 'disp_x disp_y'
    eigenstrain_names = eigenstrain
  [../]
  [./stress_3]
    type = ComputeLinearElasticStress
    base_name = phase3
  [../]
  [./eigenstrain_3]
    type = ComputeEigenstrain
    base_name = phase3
    eigen_base = '-0.05'
    eigenstrain_name = eigenstrain
  [../]

  # switching functions
  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]
  [./switching3]
    type = SwitchingFunctionMaterial
    function_name = h3
    eta = eta3
    h_order = SIMPLE
  [../]

  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2 eta3'
  [../]

  # chemical free energies
  [./chemical_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = Fc1
    expression = '4*c^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = Fc2
    expression = '(c-0.9)^2-0.4'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_3]
    type = DerivativeParsedMaterial
    property_name = Fc3
    expression = '(c-0.9)^2-0.5'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # elastic free energies
  [./elastic_free_energy_1]
    type = ElasticEnergyMaterial
    base_name = phase1
    f_name = Fe1
    derivative_order = 2
    args = 'c' # should be empty
  [../]
  [./elastic_free_energy_2]
    type = ElasticEnergyMaterial
    base_name = phase2
    f_name = Fe2
    derivative_order = 2
    args = 'c' # should be empty
  [../]
  [./elastic_free_energy_3]
    type = ElasticEnergyMaterial
    base_name = phase3
    f_name = Fe3
    derivative_order = 2
    args = 'c' # should be empty
  [../]

  # phase free energies (chemical + elastic)
  [./phase_free_energy_1]
    type = DerivativeSumMaterial
    property_name = F1
    sum_materials = 'Fc1 Fe1'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./phase_free_energy_2]
    type = DerivativeSumMaterial
    property_name = F2
    sum_materials = 'Fc2 Fe2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./phase_free_energy_3]
    type = DerivativeSumMaterial
    property_name = F3
    sum_materials = 'Fc3 Fe3'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # global free energy
  [./free_energy]
    type = DerivativeMultiPhaseMaterial
    f_name = F
    fi_names = 'F1  F2  F3'
    hi_names = 'h1  h2  h3'
    etas     = 'eta1 eta2 eta3'
    coupled_variables = 'c'
    W = 3
  [../]

  # Generate the global stress from the phase stresses
  [./global_stress]
    type = MultiPhaseStressMaterial
    phase_base = 'phase1 phase2 phase3'
    h          = 'h1     h2     h3'
  [../]
[]

[BCs]
  # the boundary conditions on the displacement enforce periodicity
  # at zero total shear and constant volume
  [./bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom'
    value = 0
  [../]
  [./top_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'top'
    value = 0
  [../]
  [./left_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0
  [../]
  [./right_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'right'
    value = 0
  [../]

  [./Periodic]
    [./disp_x]
      auto_direction = 'y'
    [../]
    [./disp_y]
      auto_direction = 'x'
    [../]

    # all other phase field variables are fully periodic
    [./c]
      auto_direction = 'x y'
    [../]
    [./eta1]
      auto_direction = 'x y'
    [../]
    [./eta2]
      auto_direction = 'x y'
    [../]
    [./eta3]
      auto_direction = 'x y'
    [../]
    [./lambda]
      auto_direction = 'x y'
    [../]
  [../]
[]

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

# We monitor the total free energy and the total solute concentration (should be constant)
[Postprocessors]
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
  [../]
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      ilu'

  l_max_its = 30
  nl_max_its = 10
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  num_steps = 200

  [./TimeStepper]
    type = SolutionTimeAdaptiveDT
    dt = 0.1
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
  [./table]
    type = CSV
    delimiter = ' '
  [../]
[]

[Debug]
  # show_var_residual_norms = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialAnisotropyAntitrap.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 15
  ny = 15
  xmin = -2
  xmax = 2
  ymin = -2
  ymax = 2
[]

[GlobalParams]
  radius = 1.0
  int_width = 0.8
  x1 = 0
  y1 = 0
  enable_jit = true
  derivative_order = 2
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
[]

[AuxKernels]
  [./bnds]
    type = BndsCalcAux
    variable = bnds
    v = 'etaa0 etab0'
  [../]
[]

[ICs]
  [./w]
    type = SmoothCircleIC
    variable = w
    outvalue = -4.0
    invalue = 0.0
  [../]
  [./etaa0]
    type = SmoothCircleIC
    variable = etaa0
    #Solid phase
    outvalue = 0.0
    invalue = 1.0
  [../]
  [./etab0]
    type = SmoothCircleIC
    variable = etab0
    #Liquid phase
    outvalue = 1.0
    invalue = 0.0
  [../]
[]

[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0'
    gamma_names = 'gab'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etab0 w'
  [../]
  [./ACa0_int1]
    type = ACInterface2DMultiPhase1
    variable = etaa0
    etas = 'etab0'
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
  [../]
  [./ACa0_int2]
    type = ACInterface2DMultiPhase2
    variable = etaa0
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0'
    gamma_names = 'gab'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etaa0 w'
  [../]
  [./ACb0_int1]
    type = ACInterface2DMultiPhase1
    variable = etab0
    etas = 'etaa0'
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
  [../]
  [./ACb0_int2]
    type = ACInterface2DMultiPhase2
    variable = etab0
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
    coupled_variables = '' # in this case chi (the susceptibility) is simply a constant
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
    args = ''
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etaa0dot_int]
    type = AntitrappingCurrent
    variable = w
    v = etaa0
    f_name = rhodiff
  [../]
  [./coupled_etab0dot_int]
    type = AntitrappingCurrent
    variable = w
    v = etab0
    f_name = rhodiff
  [../]
[]

[Materials]
  [./ha]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0'
    phase_etas = 'etaa0'
  [../]
  [./hb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0'
    phase_etas = 'etab0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegab
    material_property_names = 'Vm kb cbeq'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq'
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
  [../]
  [./int]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhodiff
    material_property_names = 'rhoa rhob'
    constant_names = 'int_width'
    constant_expressions = '0.8'
    expression = 'int_width*(rhob-rhoa)'
  [../]
  [./kappaa]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
    etaa = etaa0
    etab = etab0
  [../]
  [./kappab]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
    etaa = etab0
    etab = etaa0
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'L   D    chi  Vm   ka    caeq kb    cbeq  gab mu'
    prop_values = '1.0 1.0  0.1  1.0  10.0  0.1  10.0  0.9   4.5 10.0'
  [../]
  [./Mobility]
    type = ParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  lu           1'
  l_tol = 1.0e-3
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1e-8
  num_steps = 3
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.001
  [../]
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/MultiPhase/acmultiinterface.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 20
  ny = 10
  nz = 0
  xmin = -10
  xmax = 10
  ymin = -5
  ymax = 5
  elem_type = QUAD4
[]

[Variables]
  [./eta1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = -3.5
      y1 =  0.0
      radius = 4.0
      invalue = 0.9
      outvalue = 0.1
      int_width = 2.0
    [../]
  [../]
  [./eta2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = SmoothCircleIC
      x1 =  3.5
      y1 =  0.0
      radius = 4.0
      invalue = 0.9
      outvalue = 0.1
      int_width = 2.0
    [../]
  [../]
  [./eta3]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = SpecifiedSmoothCircleIC
      x_positions = '-4.0 4.0'
      y_positions = ' 0.0 0.0'
      z_positions = ' 0.0 0.0'
      radii = '4.0 4.0'
      invalue = 0.1
      outvalue = 0.9
      int_width = 2.0
    [../]
  [../]

  [./lambda]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1.0
  [../]
[]

[Kernels]
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    coupled_variables = 'eta2 eta3'
    mob_name = L1
    f_name = F
  [../]
  [./ACInterface1]
    type = ACMultiInterface
    variable = eta1
    etas = 'eta1 eta2 eta3'
    mob_name = L1
    kappa_names = 'kappa11 kappa12 kappa13'
  [../]
  [./lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name   = h1
    lambda = lambda
  [../]

  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    coupled_variables = 'eta1 eta3'
    mob_name = L2
    f_name = F
  [../]
  [./ACInterface2]
    type = ACMultiInterface
    variable = eta2
    etas = 'eta1 eta2 eta3'
    mob_name = L2
    kappa_names = 'kappa21 kappa22 kappa23'
  [../]
  [./lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name   = h2
    lambda = lambda
  [../]

  [./deta3dt]
    type = TimeDerivative
    variable = eta3
  [../]
  [./ACBulk3]
    type = AllenCahn
    variable = eta3
    coupled_variables = 'eta1 eta2'
    mob_name = L3
    f_name = F
  [../]
  [./ACInterface3]
    type = ACMultiInterface
    variable = eta3
    etas = 'eta1 eta2 eta3'
    mob_name = L3
    kappa_names = 'kappa31 kappa32 kappa33'
  [../]
  [./lagrange3]
    type = SwitchingFunctionConstraintEta
    variable = eta3
    h_name   = h3
    lambda = lambda
  [../]

  [./lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
    epsilon = 0
  [../]
[]

[BCs]
  [./Periodic]
    [./All]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Materials]
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'Fx  L1 L2 L3  kappa11 kappa12 kappa13 kappa21 kappa22 kappa23 kappa31 kappa32 kappa33'
    prop_values = '0   1  1  1   1       1       1       1       1       1       1       1       1      '
  [../]

  [./etasummat]
    type = ParsedMaterial
    property_name = etasum
    coupled_variables = 'eta1 eta2 eta3'
    material_property_names = 'h1 h2 h3'
    expression = 'h1+h2+h3'
  [../]

  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]
  [./switching3]
    type = SwitchingFunctionMaterial
    function_name = h3
    eta = eta3
    h_order = SIMPLE
  [../]

  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2 eta3'
  [../]

  [./free_energy]
    type = DerivativeMultiPhaseMaterial
    property_name = F
    # we use a constant free energy (GeneriConstantmaterial property Fx)
    fi_names = 'Fx  Fx  Fx'
    hi_names = 'h1  h2  h3'
    etas     = 'eta1 eta2 eta3'
    # the free energy is given by the MultiBarrierFunctionMaterial only
    W = 1
    derivative_order = 2
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = 'bdf2'

  solve_type = 'PJFNK'
  #petsc_options = '-snes_ksp -snes_ksp_ew'
  #petsc_options = '-ksp_monitor_snes_lg-snes_ksp_ew'
  #petsc_options_iname = '-ksp_gmres_restart'
  #petsc_options_value = '1000              '

  l_max_its = 15
  l_tol = 1.0e-6

  nl_max_its = 50
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1.0e-10

  start_time = 0.0
  num_steps = 2
  dt = 0.2
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/combined/examples/optimization/three_materials.i)

vol_frac = 0.4
cost_frac = 0.3

power = 4

E0 = 1.0e-6
E1 = 0.2
E2 = 0.6
E3 = 1.0

rho0 = 1.0e-6
rho1 = 0.4
rho2 = 0.7
rho3 = 1.0

C0 = 1.0e-6
C1 = 0.5
C2 = 0.8
C3 = 1.0

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 50
    ny = 50
    xmin = 0
    xmax = 50
    ymin = 0
    ymax = 50
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push_left]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push_left
    coord = '25 0 0'
  []
  [push_center]
    type = ExtraNodesetGenerator
    input = push_left
    new_boundary = push_center
    coord = '50 0 0'
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]

[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cost]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]

[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []

[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-3
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${E0}-${E1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${E0}-A1*${rho0}^${power}; E1:=A1*mat_den^${power}+B1; "
                 "A2:=(${E1}-${E2})/(${rho1}^${power}-${rho2}^${power}); "
                 "B2:=${E1}-A2*${rho1}^${power}; E2:=A2*mat_den^${power}+B2; "
                 "A3:=(${E2}-${E3})/(${rho2}^${power}-${rho3}^${power}); "
                 "B3:=${E2}-A3*${rho2}^${power}; E3:=A3*mat_den^${power}+B3; "
                 "if(mat_den<${rho1},E1,if(mat_den<${rho2},E2,E3))"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []

  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^(1/${power})-${rho1}^(1/${power})); "
                 "B1:=${C0}-A1*${rho0}^(1/${power}); C1:=A1*mat_den^(1/${power})+B1; "
                 "A2:=(${C1}-${C2})/(${rho1}^(1/${power})-${rho2}^(1/${power})); "
                 "B2:=${C1}-A2*${rho1}^(1/${power}); C2:=A2*mat_den^(1/${power})+B2; "
                 "A3:=(${C2}-${C3})/(${rho2}^(1/${power})-${rho3}^(1/${power})); "
                 "B3:=${C2}-A3*${rho2}^(1/${power}); C3:=A3*mat_den^(1/${power})+B3; "
                 "if(mat_den<${rho1},C1,if(mat_den<${rho2},C2,C3))"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
  []
  [CostDensity]
    type = ParsedMaterial
    property_name = CostDensity
    coupled_variables = 'mat_den Cost'
    expression = 'mat_den*Cost'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = Cost_mat
    outputs = 'exodus'
  []
[]

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

[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 3
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 3
    weights = linear
    prop_name = cost_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdateTwoConstraints
    # This is
    density_sensitivity = Dc
    cost_density_sensitivity = Cc
    cost = Cost
    cost_fraction = ${cost_frac}
    design_density = mat_den
    volume_fraction = ${vol_frac}
    bisection_lower_bound = 0
    bisection_upper_bound = 1.0e16 # 100
    bisection_move = 0.05
    adaptive_move = true
    relative_tolerance = 1.0e-3
    execute_on = TIMESTEP_BEGIN
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Cc
  [calc_sense_cost]
    type = SensitivityFilter
    density_sensitivity = Cc
    design_density = mat_den
    filter_UO = rad_avg_cost
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-10
  dt = 1.0
  num_steps = 40
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralMaterialProperty
    mat_prop = CostDensity
  []
  [cost_frac]
    type = ParsedPostprocessor
    expression = 'cost / mesh_volume'
    pp_names = 'cost mesh_volume'
  []

[]

(modules/combined/test/tests/phase_field_fracture/crack2d_vol_dev.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = strain_vol_dev
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/anisotropic_interfaces/GrandPotentialPlanarGrowth.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmin = -2
  xmax = 2
  ymin = -2
  ymax = 2
  uniform_refine = 2
[]

[GlobalParams]
  x1 = -2
  y1 = -2
  x2 = 2
  y2 = -1.5
  derivative_order = 2
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
  #Temperature
  [./T]
  [../]
[]

[AuxKernels]
  [./bnds]
    type = BndsCalcAux
    variable = bnds
    v = 'etaa0 etab0'
  [../]
  [./T]
    type = FunctionAux
    function = 95.0+2.0*(y-1.0*t)
    variable = T
    execute_on = 'initial timestep_begin'
  [../]
[]

[ICs]
  [./w]
    type = BoundingBoxIC
    variable = w
    # note w = A*(c-cleq), A = 1.0, cleq = 0.0 ,i.e., w = c (in the matrix/liquid phase)
    outside = -4.0
    inside = 0.0
  [../]
  [./etaa0]
    type = BoundingBoxIC
    variable = etaa0
    #Solid phase
    outside = 0.0
    inside = 1.0
  [../]
  [./etab0]
    type = BoundingBoxIC
    variable = etab0
    #Liquid phase
    outside = 1.0
    inside = 0.0
  [../]
[]


[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0'
    gamma_names = 'gab'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etab0 w'
  [../]
  [./ACa0_int1]
    type = ACInterface2DMultiPhase1
    variable = etaa0
    etas = 'etab0'
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
  [../]
  [./ACa0_int2]
    type = ACInterface2DMultiPhase2
    variable = etaa0
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0'
    gamma_names = 'gab'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etaa0 w'
  [../]
  [./ACb0_int1]
    type = ACInterface2DMultiPhase1
    variable = etab0
    etas = 'etaa0'
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
  [../]
  [./ACb0_int2]
    type = ACInterface2DMultiPhase2
    variable = etab0
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
[]


[Materials]
  [./ha]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0'
    phase_etas = 'etaa0'
  [../]
  [./hb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0'
    phase_etas = 'etab0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w T'
    property_name = omegab
    material_property_names = 'Vm kb cbeq S Tm'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq-S*(T-Tm)'
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
  [../]
  [./kappaa]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
    etaa = etaa0
    etab = etab0
    outputs = exodus
    output_properties = 'kappaa'
  [../]
  [./kappab]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
    etaa = etab0
    etab = etaa0
    outputs = exodus
    output_properties = 'kappab'
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'L   D    chi  Vm   ka    caeq kb    cbeq  gab mu   S   Tm'
    prop_values = '1.0 1.0  0.1  1.0  10.0  0.1  10.0  0.9   4.5 10.0 1.0 100.0'
  [../]
  [./Mobility]
    type = ParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart'
  petsc_options_value = 'hypre    boomeramg      31'
  l_tol = 1.0e-3
  l_max_its = 30
  nl_max_its = 15
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1e-8
  end_time = 2.0
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.0005
    cutback_factor = 0.7
    growth_factor = 1.2
  [../]
[]

[Adaptivity]
 initial_steps = 3
 max_h_level = 3
 initial_marker = err_eta
 marker = err_bnds
[./Markers]
   [./err_eta]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_eta
   [../]
   [./err_bnds]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_bnds
   [../]
 [../]
 [./Indicators]
   [./ind_eta]
     type = GradientJumpIndicator
     variable = etaa0
    [../]
    [./ind_bnds]
      type = GradientJumpIndicator
      variable = bnds
   [../]
 [../]
[]

[Outputs]
  time_step_interval = 10
  exodus = true
[]

(modules/combined/test/tests/optimization/invOpt_nonlinear/homogeneous_forward.i)

[Executioner]
  type = Steady
  solve_type = NEWTON
  line_search = none
  #nl_forced_its = 1
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Mesh]
[]

[Variables]
  [T]
  []
[]

[AuxVariables]
  [forwardT]
  []
  [_dDdTgradT]
    order  = CONSTANT
    family = MONOMIAL_VEC
  []
[]

[Kernels]
  [heat_conduction]
    type = ADHeatConduction
    thermal_conductivity = 'linearized_conductivity'
    variable = T
  []
  [heat_source]
    type = ADMatHeatSource
    material_property = 'volumetric_heat'
    variable = T
  []
  [advection]
    type = ConservativeAdvection
    velocity = _dDdTgradT
    variable = T
    upwinding_type = full  #Full upwinding gives somewhat better results
  []
[]

[AuxKernels]
  [_dDdTgradT]
    type = ADFunctorElementalGradientAux
    functor = forwardT
    variable = _dDdTgradT
    factor_matprop = '_dDdT'
  []
[]

[Materials]
  [LinearizedConductivity]
    type = ADParsedMaterial
    f_name = 'linearized_conductivity'
    function = '10+500*forwardT'
    args = 'forwardT'
  []
  [_dDdT]
    type = ADParsedMaterial
    f_name = '_dDdT' # "_" represents negation
    function = '-500'
    args = 'forwardT'
  []
  [volumetric_heat]
    type = ADGenericFunctionMaterial
    prop_names = 'volumetric_heat'
    prop_values = 'volumetric_heat_func'
  []
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/heat_source'
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = T
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = T
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = T
    boundary = bottom
    value = 0
  []
  [top]
    type = DirichletBC
    variable = T
    boundary = top
    value = 0
  []
[]

[Reporters]
  [measurement_locations]
    type = OptimizationData
    variable = T
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'heat_source'
    real_vector_values = '0' # Dummy
  []
[]

[Outputs]
  console = false
[]

(modules/combined/examples/publications/rapid_dev/fig8.i)

#
# Fig. 8 input for 10.1016/j.commatsci.2017.02.017
# D. Schwen et al./Computational Materials Science 132 (2017) 36-45
# Two growing particles with differnet anisotropic Eigenstrains
#

[Mesh]
  [./gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 80
    ny = 40
    xmin = -20
    xmax = 20
    ymin = 0
    ymax = 20
    elem_type = QUAD4
  [../]
  [./cnode]
    type = ExtraNodesetGenerator
    input = gen
    coord = '0.0 0.0'
    new_boundary = 100
    tolerance = 0.1
  [../]
[]

[GlobalParams]
  # CahnHilliard needs the third derivatives
  derivative_order = 3
  enable_jit = true
  displacements = 'disp_x disp_y'
  int_width = 1
[]

# AuxVars to compute the free energy density for outputting
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    additional_free_energy = cross_energy
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
  [./cross_terms]
    type = CrossTermGradientFreeEnergy
    variable = cross_energy
    interfacial_vars = 'eta1 eta2 eta3'
    kappa_names = 'kappa11 kappa12 kappa13
                   kappa21 kappa22 kappa23
                   kappa31 kappa32 kappa33'
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

# particle x positions and radius
P1X=8
P2X=-4
PR=2

[Variables]
  # Solute concentration variable
  [./c]
    [./InitialCondition]
      type = SpecifiedSmoothCircleIC
      x_positions = '${P1X} ${P2X}'
      y_positions = '0 0'
      z_positions = '0 0'
      radii = '${PR} ${PR}'
      outvalue = 0.5
      invalue = 0.9
    [../]
  [../]
  [./w]
  [../]

  # Order parameter for the Matrix
  [./eta1]
    [./InitialCondition]
      type = SpecifiedSmoothCircleIC
      x_positions = '${P1X} ${P2X}'
      y_positions = '0 0'
      z_positions = '0 0'
      radii = '${PR} ${PR}'
      outvalue = 1.0
      invalue = 0.0
    [../]
  [../]
  # Order parameters for the 2 different inclusion orientations
  [./eta2]
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = ${P2X}
      y1 = 0
      radius = ${PR}
      invalue = 1.0
      outvalue = 0.0
    [../]
  [../]
  [./eta3]
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = ${P1X}
      y1 = 0
      radius = ${PR}
      invalue = 1.0
      outvalue = 0.0
    [../]
  [../]

  # Lagrange-multiplier
  [./lambda]
    initial_condition = 1.0
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./all]
        add_variables = true
        strain = SMALL
        eigenstrain_names = eigenstrain
      [../]
    [../]
  [../]
[]

[Kernels]
  # Split Cahn-Hilliard kernels
  [./c_res]
    type = SplitCHParsed
    variable = c
    f_name = F
    args = 'eta1 eta2 eta3'
    kappa_name = kappa_c
    w = w
  [../]
  [./wres]
    type = SplitCHWRes
    variable = w
    mob_name = M
  [../]
  [./time]
    type = CoupledTimeDerivative
    variable = w
    v = c
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 1
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    args = 'eta2 eta3 c'
    mob_name = L1
    f_name = F
  [../]
  [./ACInterface1]
    type = ACMultiInterface
    variable = eta1
    etas = 'eta1 eta2 eta3'
    mob_name = L1
    kappa_names = 'kappa11 kappa12 kappa13'
  [../]
  [./lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name   = h1
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 2
  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    args = 'eta1 eta3 c'
    mob_name = L2
    f_name = F
  [../]
  [./ACInterface2]
    type = ACMultiInterface
    variable = eta2
    etas = 'eta1 eta2 eta3'
    mob_name = L2
    kappa_names = 'kappa21 kappa22 kappa23'
  [../]
  [./lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name   = h2
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 3
  [./deta3dt]
    type = TimeDerivative
    variable = eta3
  [../]
  [./ACBulk3]
    type = AllenCahn
    variable = eta3
    args = 'eta1 eta2 c'
    mob_name = L3
    f_name = F
  [../]
  [./ACInterface3]
    type = ACMultiInterface
    variable = eta3
    etas = 'eta1 eta2 eta3'
    mob_name = L3
    kappa_names = 'kappa31 kappa32 kappa33'
  [../]
  [./lagrange3]
    type = SwitchingFunctionConstraintEta
    variable = eta3
    h_name   = h3
    lambda = lambda
  [../]

  # Lagrange-multiplier constraint kernel for lambda
  [./lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
    epsilon = 1e-6
  [../]
[]

[Materials]
  # declare a few constants, such as mobilities (L,M) and interface gradient prefactors (kappa*)
  [./consts]
    type = GenericConstantMaterial
    block = 0
    prop_names  = 'M   kappa_c  L1 L2 L3  kappa11 kappa12 kappa13 kappa21 kappa22 kappa23 kappa31 kappa32 kappa33'
    prop_values = '0.2 0.5      1  1  1   2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00   '
  [../]

  # We use this to output the level of constraint enforcement
  # ideally it should be 0 everywhere, if the constraint is fully enforced
  [./etasummat]
    type = ParsedMaterial
    property_name = etasum
    coupled_variables = 'eta1 eta2 eta3'
    material_property_names = 'h1 h2 h3'
    expression = 'h1+h2+h3-1'
    outputs = exodus
  [../]

  # This parsed material creates a single property for visualization purposes.
  # It will be 0 for phase 1, -1 for phase 2, and 1 for phase 3
  [./phasemap]
    type = ParsedMaterial
    property_name = phase
    coupled_variables = 'eta2 eta3'
    expression = 'if(eta3>0.5,1,0)-if(eta2>0.5,1,0)'
    outputs = exodus
  [../]

  # global mechanical properties
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '400 400'
    fill_method = symmetric_isotropic
  [../]
  [./stress]
    type = ComputeLinearElasticStress
  [../]

  # eigenstrain
  [./eigenstrain_2]
    type = GenericConstantRankTwoTensor
    tensor_name = s2
    tensor_values = '0 -0.05 0  0 0 0'
  [../]
  [./eigenstrain_3]
    type = GenericConstantRankTwoTensor
    tensor_name = s3
    tensor_values =  '-0.05 0 0  0 0 0'
  [../]
  [./eigenstrain]
    type = CompositeEigenstrain
    weights = 'h2 h3'
    tensors = 's2 s3'
    args = 'eta2 eta3'
    eigenstrain_name = eigenstrain
  [../]

  # switching functions
  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]
  [./switching3]
    type = SwitchingFunctionMaterial
    function_name = h3
    eta = eta3
    h_order = SIMPLE
  [../]

  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2 eta3'
  [../]

  # chemical free energies
  [./chemical_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = Fc1
    expression = '4*c^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = Fc2
    expression = '(c-0.9)^2-0.4'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_3]
    type = DerivativeParsedMaterial
    property_name = Fc3
    expression = '(c-0.9)^2-0.5'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # global chemical free energy
  [./chemical_free_energy]
    type = DerivativeMultiPhaseMaterial
    f_name = Fc
    fi_names = 'Fc1  Fc2  Fc3'
    hi_names = 'h1  h2  h3'
    etas     = 'eta1 eta2 eta3'
    coupled_variables = 'c'
    W = 3
  [../]

  # global elastic free energy
  [./elastic_free_energy]
    type = ElasticEnergyMaterial
    f_name = Fe
    args = 'eta2 eta3'
    outputs = exodus
    output_properties = Fe
    derivative_order = 2
  [../]

  # Penalize phase 2 and 3 coexistence
  [./multi_phase_penalty]
    type = DerivativeParsedMaterial
    property_name = Fp
    expression = '50*(eta2*eta3)^2'
    coupled_variables = 'eta2 eta3'
    derivative_order = 2
    outputs = exodus
    output_properties = Fp
  [../]

  # free energy
  [./free_energy]
    type = DerivativeSumMaterial
    property_name = F
    sum_materials = 'Fc Fe Fp'
    coupled_variables = 'c eta1 eta2 eta3'
    derivative_order = 2
  [../]
[]

[BCs]
  # fix center point location
  [./centerfix_x]
    type = DirichletBC
    boundary = 100
    variable = disp_x
    value = 0
  [../]

  # fix side point x coordinate to inhibit rotation
  [./angularfix]
    type = DirichletBC
    boundary = bottom
    variable = disp_y
    value = 0
  [../]
[]

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

# We monitor the total free energy and the total solute concentration (should be constant)
[Postprocessors]
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu'

  l_max_its = 30
  nl_max_its = 10
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  end_time = 12.0

  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 8
    iteration_window = 1
    dt = 0.01
  [../]
[]

[Outputs]
  print_linear_residuals = false
  execute_on = 'INITIAL TIMESTEP_END'
  exodus = true
  [./table]
    type = CSV
    delimiter = ' '
  [../]
[]

[Debug]
  # show_var_residual_norms = true
[]

(modules/combined/examples/publications/rapid_dev/fig7b.i)

#
# Fig. 7 input for 10.1016/j.commatsci.2017.02.017
# D. Schwen et al./Computational Materials Science 132 (2017) 36-45
# Dashed black curve (2)
# Eigenstrain is globally applied. Single global elastic free energies.
# Supply the RADIUS parameter (10-35) on the command line to generate data
# for all curves in the plot.
#

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 32
  xmin = 0
  xmax = 100
  second_order = true
[]

[Problem]
  coord_type = RSPHERICAL
[]

[GlobalParams]
  displacements = 'disp_r'
[]

[Functions]
  [./diff]
    type = ParsedFunction
    expression = '${RADIUS}-pos_c'
    symbol_names = pos_c
    symbol_values = pos_c
  [../]
[]

# AuxVars to compute the free energy density for outputting
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

[Variables]
  # Solute concentration variable
  [./c]
    [./InitialCondition]
      type = SmoothCircleIC
      invalue = 1
      outvalue = 0
      x1 = 0
      y1 = 0
      radius = ${RADIUS}
      int_width = 3
    [../]
  [../]
  [./w]
  [../]

  # Phase order parameter
  [./eta]
    [./InitialCondition]
      type = SmoothCircleIC
      invalue = 1
      outvalue = 0
      x1 = 0
      y1 = 0
      radius = ${RADIUS}
      int_width = 3
    [../]
  [../]

  [./Fe_fit]
    order = SECOND
  [../]
[]

[Modules/TensorMechanics/Master/all]
  add_variables = true
  eigenstrain_names = eigenstrain
[]

[Kernels]
  # Split Cahn-Hilliard kernels
  [./c_res]
    type = SplitCHParsed
    variable = c
    f_name = F
    args = 'eta'
    kappa_name = kappa_c
    w = w
  [../]
  [./wres]
    type = SplitCHWRes
    variable = w
    mob_name = M
  [../]
  [./time]
    type = CoupledTimeDerivative
    variable = w
    v = c
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 1
  [./detadt]
    type = TimeDerivative
    variable = eta
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta
    args = 'c'
    mob_name = L
    f_name = F
  [../]
  [./ACInterface]
    type = ACInterface
    variable = eta
    mob_name = L
    kappa_name = kappa_eta
  [../]

  [./Fe]
    type = MaterialPropertyValue
    prop_name = Fe
    variable = Fe_fit
  [../]

  [./autoadjust]
    type = MaskedBodyForce
    variable = w
    function = diff
    mask = mask
  [../]
[]

[Materials]
  # declare a few constants, such as mobilities (L,M) and interface gradient prefactors (kappa*)
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'M   L   kappa_c kappa_eta'
    prop_values = '1.0 1.0 0.5     1'
  [../]

  # forcing function mask
  [./mask]
    type = ParsedMaterial
    property_name = mask
    expression = grad/dt
    material_property_names = 'grad dt'
  [../]
  [./grad]
    type = VariableGradientMaterial
    variable = c
    prop = grad
  [../]
  [./time]
    type = TimeStepMaterial
  [../]

  # global mechanical properties
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '1 1'
    fill_method = symmetric_isotropic
  [../]
  [./stress]
    type = ComputeLinearElasticStress
  [../]

  # eigenstrain as a function of phase
  [./eigenstrain]
    type = ComputeVariableEigenstrain
    eigen_base = '0.05 0.05 0.05 0 0 0'
    prefactor = h
    args = eta
    eigenstrain_name = eigenstrain
  [../]

  # switching functions
  [./switching]
    type = SwitchingFunctionMaterial
    function_name = h
    eta = eta
    h_order = SIMPLE
  [../]
  [./barrier]
    type = BarrierFunctionMaterial
    eta = eta
  [../]

  # chemical free energies
  [./chemical_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = Fc1
    expression = 'c^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = Fc2
    expression = '(1-c)^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # global chemical free energy
  [./chemical_free_energy]
    type = DerivativeTwoPhaseMaterial
    f_name = Fc
    fa_name = Fc1
    fb_name = Fc2
    eta = eta
    args = 'c'
    W = 4
  [../]

  # global elastic free energy
  [./elastic_free_energy]
    type = ElasticEnergyMaterial
    f_name = Fe
    args = 'eta'
    output_properties = Fe
    derivative_order = 2
  [../]

  # free energy
  [./free_energy]
    type = DerivativeSumMaterial
    property_name = F
    sum_materials = 'Fc Fe'
    coupled_variables = 'c eta'
    derivative_order = 2
  [../]
[]

[BCs]
  [./left_r]
    type = DirichletBC
    variable = disp_r
    boundary = 'left'
    value = 0
  [../]
[]

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

# We monitor the total free energy and the total solute concentration (should be constant)
[Postprocessors]
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./pos_c]
    type = FindValueOnLine
    start_point = '0 0 0'
    end_point = '100 0 0'
    v = c
    target = 0.582
    tol = 1e-8
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./pos_eta]
    type = FindValueOnLine
    start_point = '0 0 0'
    end_point = '100 0 0'
    v = eta
    target = 0.5
    tol = 1e-8
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./c_min]
    type = ElementExtremeValue
    value_type = min
    variable = c
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
[]

[VectorPostprocessors]
  [./line]
    type = LineValueSampler
    variable = 'Fe_fit c w'
    start_point = '0 0 0'
    end_point =   '100 0 0'
    num_points = 5000
    sort_by = x
    outputs = vpp
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu'

  l_max_its = 30
  nl_max_its = 15
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 2.0e-9
  start_time = 0.0
  end_time = 100000.0

  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 8
    iteration_window = 1
    dt = 1
  [../]

  [./Adaptivity]
    initial_adaptivity = 5
    interval = 10
    max_h_level = 5
    refine_fraction = 0.9
    coarsen_fraction = 0.1
  [../]
[]

[Outputs]
  print_linear_residuals = false
  perf_graph = true
  execute_on = 'INITIAL TIMESTEP_END'
  [./table]
    type = CSV
    delimiter = ' '
    file_base = radius_${RADIUS}/eigenstrain_pp
  [../]
  [./vpp]
    type = CSV
    delimiter = ' '
    sync_times = '10 50 100 500 1000 5000 10000 50000 100000'
    sync_only = true
    time_data = true
    file_base = radius_${RADIUS}/eigenstrain_vpp
  [../]
[]

(modules/optimization/examples/simpleTransient/nonlinear_forward_and_adjoint.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = -1
    xmax = 1
    ymin = -1
    ymax = 1
  []
[]

[Variables]
  [u]
  []
[]

[VectorPostprocessors]
  [src_values]
    type = CSVReader
    csv_file = source_params.csv
    header = true
    outputs = none
  []
[]

[ICs]
  [initial]
    type = FunctionIC
    variable = u
    function = exact
  []
[]

[Kernels]
  [dt]
    type = ADTimeDerivative
    variable = u
  []
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = D
  []
  [src]
    type = ADBodyForce
    variable = u
    function = source
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u
    boundary = 'left right top bottom'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = ADParsedMaterial
    property_name = D
    expression = '1 + u'
    coupled_variables = u
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = '2*exp(-2.0*(x - sin(2*pi*t))^2)*exp(-2.0*(y - cos(2*pi*t))^2)*cos((1/2)*x*pi)*cos((1/2)*y*pi)/pi'
  []
  [source]
    type = NearestReporterCoordinatesFunction
    x_coord_name = src_values/coordx
    y_coord_name = src_values/coordy
    time_name = src_values/time
    value_name = src_values/values
  []
[]

[Executioner]
  type = TransientAndAdjoint
  forward_system = nl0
  adjoint_system = adjoint

  num_steps = 100
  end_time = 1

  nl_rel_tol = 1e-12
  l_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Reporters]
  [measured_data]
    type = OptimizationData
    measurement_file = mms_data.csv
    file_xcoord = x
    file_ycoord = y
    file_zcoord = z
    file_time = t
    file_value = u
    variable = u
    execute_on = timestep_end
    outputs = none
  []
[]

[Postprocessors]
  [topRight_pp]
    type = PointValue
    point = '0.5 0.5 0'
    variable = u
    execute_on = TIMESTEP_END
  []
  [bottomRight_pp]
    type = PointValue
    point = '-0.5 0.5 0'
    variable = u
    execute_on = TIMESTEP_END
  []
  [bottomLeft_pp]
    type = PointValue
    point = '-0.5 -0.5 0'
    variable = u
    execute_on = TIMESTEP_END
  []
  [topLeft_pp]
    type = PointValue
    point = '0.5 -0.5 0'
    variable = u
    execute_on = TIMESTEP_END
  []
[]

[Outputs]
  csv = true
  console = false
[]

[Problem]
  nl_sys_names = 'nl0 adjoint'
  kernel_coverage_check = false
[]

[Variables]
  [u_adjoint]
    solver_sys = adjoint
    outputs = none
  []
[]

[DiracKernels]
  [misfit]
    type = ReporterTimePointSource
    variable = u_adjoint
    value_name = measured_data/misfit_values
    x_coord_name = measured_data/measurement_xcoord
    y_coord_name = measured_data/measurement_ycoord
    z_coord_name = measured_data/measurement_zcoord
    time_name = measured_data/measurement_time
  []
[]

[VectorPostprocessors]
  [adjoint]
    type = ElementOptimizationSourceFunctionInnerProduct
    variable = u_adjoint
    function = source
    execute_on = ADJOINT_TIMESTEP_END
    outputs = none
  []
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_iso_with_pressure.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco fracture_pressure'
    prop_values = '1e-3 0.04 1e-4 1e-3'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    I_name = 'indicator_function'
    F_name = 'local_fracture_energy'
    decomposition_type = strain_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./indicator_function]
    type = DerivativeParsedMaterial
    property_name = indicator_function
    coupled_variables = 'c'
    expression = 'c'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_iso_wo_time.i)

#This input does not add time derivative kernel for phase field equation
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[Variables]
  [./c]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
  [./ACBulk]
    type = AllenCahn
    variable = c
    f_name = F
  [../]

  [./ACInterface]
    type = ACInterface
    variable = c
    kappa_name = kappa_op
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./elastic]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = strain_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(test/tests/kernels/ad_mat_reaction/ad_mat_reaction.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
  []
[]

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

[Materials]
  [ad_prop]
    type = ADParsedMaterial
    expression = '-log(3)*log(3)'
    property_name = rxn_prop
  []
[]

[Kernels]
  [diff]
    type = ADDiffusion
    variable = u
  []
  [reaction]
    type = ADMatReaction
    variable = u
    reaction_rate = rxn_prop
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 1
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 3
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/SoretDiffusion/split_temp.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 60
  xmax = 500
  elem_type = EDGE
[]

[GlobalParams]
  polynomial_order = 8
[]

[Variables]
  [./c]
  [../]
  [./w]
    scaling = 1.0e2
  [../]
  [./T]
    initial_condition = 1000.0
    scaling = 1.0e5
  [../]
[]

[ICs]
  [./c_IC]
    type = SmoothCircleIC
    x1 = 125.0
    y1 = 0.0
    radius = 60.0
    invalue = 1.0
    outvalue = 0.1
    int_width = 100.0
    variable = c
  [../]
[]

[Kernels]
  [./c_res]
    type = SplitCHParsed
    variable = c
    kappa_name = kappa
    w = w
    f_name = F
  [../]
  [./w_res]
    type = SplitCHWRes
    variable = w
    mob_name = M
  [../]
  [./w_res_soret]
    type = SoretDiffusion
    variable = w
    c = c
    T = T
    diff_name = D
    Q_name = Qstar
  [../]
  [./time]
    type = CoupledTimeDerivative
    variable = w
    v = c
  [../]
  [./HtCond]
    type = MatDiffusion
    variable = T
    diffusivity = thermal_conductivity
  [../]
[]

[BCs]
  [./Left_T]
    type = DirichletBC
    variable = T
    boundary = left
    value = 1000.0
  [../]

  [./Right_T]
    type = DirichletBC
    variable = T
    boundary = right
    value = 1015.0
  [../]
[]

[Materials]
  [./Copper]
    type = PFParamsPolyFreeEnergy
    block = 0
    c = c
    T = T # K
    int_width = 60.0
    length_scale = 1.0e-9
    time_scale = 1.0e-9
    D0 = 3.1e-5 # m^2/s, from Brown1980
    Em = 0.71 # in eV, from Balluffi1978 Table 2
    Ef = 1.28 # in eV, from Balluffi1978 Table 2
    surface_energy = 0.708 # Total guess
  [../]
  [./thcond]
    type = ParsedMaterial
    block = 0
    coupled_variables = 'c'
    expression = 'if(c>0.7,1e-8,4e-8)'
    property_name = thermal_conductivity
    outputs = exodus
  [../]
  [./free_energy]
    type = PolynomialFreeEnergy
    block = 0
    c = c
    derivative_order = 2
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = 'bdf2'
  solve_type = 'PJFNK'

  l_max_its = 30
  l_tol = 1.0e-4
  nl_max_its = 25
  nl_rel_tol = 1.0e-9

  num_steps = 60
  dt = 20.0
[]

[Outputs]
   exodus = true
[]

(modules/solid_mechanics/test/tests/crystal_plasticity/cp_eigenstrains/volumetric_eigenstrain_parabolic.i)

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  [cube]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 2
    ny = 2
    nz = 2
    elem_type = HEX27
  []
[]

[AuxVariables]
  [temperature]
    order = FIRST
    family = LAGRANGE
  []
  [linear_void_strain]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_total_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_total_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [pk2_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_zz]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = FINITE
  incremental = true
  add_variables = true
[]

[Functions]
  [temperature_ramp]
    type = ParsedFunction
    expression = '600.0 + t'
  []
[]

[AuxKernels]
  [temperature]
    type = FunctionAux
    variable = temperature
    function = 'temperature_ramp'
    execute_on = timestep_begin
  []
  [linear_void_strain]
    type = MaterialRealAux
    variable = linear_void_strain
    property = equivalent_linear_change
    execute_on = timestep_end
  []
  [e_total_xx]
    type = RankTwoAux
    variable = e_total_xx
    rank_two_tensor = total_lagrangian_strain
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [e_total_yy]
    type = RankTwoAux
    variable = e_total_yy
    rank_two_tensor = total_lagrangian_strain
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [e_void_xx]
    type = RankTwoAux
    variable = e_void_xx
    rank_two_tensor = void_eigenstrain
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [e_void_yy]
    type = RankTwoAux
    variable = e_void_yy
    rank_two_tensor = void_eigenstrain
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [e_void_zz]
    type = RankTwoAux
    variable = e_void_zz
    rank_two_tensor = void_eigenstrain
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [f_void_zz]
    type = RankTwoAux
    variable = f_void_zz
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [pk2_zz]
    type = RankTwoAux
    variable = pk2_zz
    rank_two_tensor = second_piola_kirchhoff_stress
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [fp_zz]
    type = RankTwoAux
    variable = fp_zz
    rank_two_tensor = plastic_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
[]

[BCs]
  [symmy]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
  [symmx]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [symmz]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0
  []
  [hold_front]
    type = DirichletBC
    variable = disp_z
    boundary = front
    value = 0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeElasticityTensorCP
    C_ijkl = '1.98e5 1.25e5 1.25e5 1.98e5 1.25e5 1.98e5 1.22e5 1.22e5 1.22e5'
    fill_method = symmetric9
  []
  [stress]
    type = ComputeMultipleCrystalPlasticityStress
    crystal_plasticity_models = 'trial_xtalpl'
    eigenstrain_names = void_eigenstrain
    tan_mod_type = exact
    line_search_method = CUT_HALF
    use_line_search = true
    maximum_substep_iteration = 5
  []

  [trial_xtalpl]
    type = CrystalPlasticityKalidindiUpdate
    number_slip_systems = 12
    slip_sys_file_name = input_slip_sys.txt
  []
  [void_eigenstrain]
    type = ComputeCrystalPlasticityVolumetricEigenstrain
    eigenstrain_name = void_eigenstrain
    deformation_gradient_name = volumetric_deformation_gradient
    mean_spherical_void_radius = void_radius
    spherical_void_number_density = void_density
  []
  [void_density]
    type = ParsedMaterial
    property_name = void_density
    coupled_variables = temperature
    expression = 'if(temperature<601.0, 1.0e6, (-1.0e6 * (temperature - 603)^2.0 + 5.0e6))'
  []
  [void_radius]
    type = ParsedMaterial
    property_name = void_radius
    coupled_variables = temperature
    expression = 'if(temperature<601.0, 1.0e-6, (-2.25e-6 * (temperature - 603)^2.0 + 1.0e-5))'
  []
[]

[Postprocessors]

  [linear_void_strain]
    type = ElementAverageValue
    variable = linear_void_strain
  []
  [e_void_xx]
    type = ElementAverageValue
    variable = e_void_xx
  []
  [e_void_yy]
    type = ElementAverageValue
    variable = e_void_yy
  []
  [e_void_zz]
    type = ElementAverageValue
    variable = e_void_zz
  []
  [f_void_zz]
    type = ElementAverageValue
    variable = f_void_zz
  []
  [void_density]
    type = ElementAverageMaterialProperty
    mat_prop = void_density
    execute_on = TIMESTEP_END
  []
  [void_radius]
    type = ElementAverageMaterialProperty
    mat_prop = void_radius
    execute_on = TIMESTEP_END
  []
  [pk2_zz]
   type = ElementAverageValue
   variable = pk2_zz
  []
  [fp_zz]
    type = ElementAverageValue
    variable = fp_zz
  []
  [e_total_xx]
    type = ElementAverageValue
    variable = e_total_xx
  []
  [e_total_yy]
    type = ElementAverageValue
    variable = e_total_yy
  []
[]

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

[Executioner]
  type = Transient
  solve_type = 'NEWTON'

  petsc_options = '-snes_converged_reason'
  petsc_options_iname = '-pc_type -pc_asm_overlap -sub_pc_type -ksp_type -ksp_gmres_restart'
  petsc_options_value = ' asm      2              lu            gmres     200'

  line_search = 'none'
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-8
  nl_forced_its = 1

  dt = 1.0
  dtmin = 1.0e-4
  end_time  = 5.0
[]

[Outputs]
  csv = true
  perf_graph = true
[]

(modules/combined/test/tests/optimization/invOpt_nonlinear/simulation.i)

[Executioner]
  type = Steady
  solve_type = NEWTON
  line_search = none
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]


[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 2
  ymax = 2
[]
[Variables]
  [forwardT]
  []
[]


[Kernels]
  [heat_conduction]
    type = ADHeatConduction
    thermal_conductivity = 'conductivity'
    variable = forwardT
  []
  [heat_source]
    type = ADMatHeatSource
    material_property = 'volumetric_heat'
    variable = forwardT
  []
[]
[Materials]
  [NonlinearConductivity]
    type = ADParsedMaterial
    f_name = 'conductivity'
    function = '10+500*forwardT'
    args = 'forwardT'
  []
  [volumetric_heat]
    type = ADGenericFunctionMaterial
    prop_names = 'volumetric_heat'
    prop_values = 'volumetric_heat_func'
  []
[]
[Functions]
  [volumetric_heat_func]
    type = ParsedFunction
    value = q
    vars = 'q'
    vals = 'heat_source_pp'
  []
[]
[Postprocessors]
  [heat_source_pp]
    type = ConstantValuePostprocessor
    value = 333
    execute_on = 'LINEAR'
  []
[]


[BCs]
  [left]
    type = NeumannBC
    variable = forwardT
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = forwardT
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = forwardT
    boundary = bottom
    value = 2
  []
  [top]
    type = DirichletBC
    variable = forwardT
    boundary = top
    value = 1
  []
[]


[Reporters]
  [measurement_locations]
    type = OptimizationData
  []
[]
[Controls]
  [parameterReceiver]
    type = ControlsReceiver
  []
[]

[VectorPostprocessors]
  [data_pt]
    type = PointValueSampler
    variable = forwardT
    points = '0.2 0.2 0
              0.8 0.6 0
              0.2 1.4 0
              0.8 1.8 0'
    sort_by = id
  []
[]

[Outputs]
  csv = true
[]

(modules/combined/test/tests/optimization/invOpt_nonlinear/forward.i)

[Executioner]
  type = Steady
  solve_type = NEWTON
  line_search = none
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Mesh]
[]

[Variables]
  [forwardT]
  []
[]

[Kernels]
  [heat_conduction]
    type = ADHeatConduction
    thermal_conductivity = 'conductivity'
    variable = forwardT
  []
  [heat_source]
    type = ADMatHeatSource
    material_property = 'volumetric_heat'
    variable = forwardT
  []
[]

[Materials]
  [NonlinearConductivity]
    type = ADParsedMaterial
    f_name = 'conductivity'
    function = '10+500*forwardT'
    args = 'forwardT'
  []
  [volumetric_heat]
    type = ADGenericFunctionMaterial
    prop_names = 'volumetric_heat'
    prop_values = 'volumetric_heat_func'
  []
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/heat_source'
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = forwardT
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = forwardT
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = forwardT
    boundary = bottom
    value = 2
  []
  [top]
    type = DirichletBC
    variable = forwardT
    boundary = top
    value = 1
  []
[]

[Reporters]
  [measurement_locations]
    type = OptimizationData
    variable = forwardT
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'heat_source'
    real_vector_values = '0' # Dummy
  []
[]

[Outputs]
  console = false
[]

(modules/phase_field/test/tests/actions/gpm_kernel.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 100
  xmin = 0
  xmax = 300
[]

[GlobalParams]
  op_num = 1
  var_name_base = eta
[]

[Variables]
  [./w]
  [../]
  [./phi]
  [../]
  [./eta0]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
[]

[ICs]
  [./IC_w]
    type = BoundingBoxIC
    variable = w
    x1 = 150
    x2 = 300
    y1 = 0
    y2 = 0
    inside = 0.1
    outside = 0
  [../]
  [./IC_phi]
    type = BoundingBoxIC
    variable = phi
    x1 = 0
    x2 = 150
    y1 = 0
    y2 = 0
    inside = 1
    outside = 0
  [../]
  [./IC_eta0]
    type = BoundingBoxIC
    variable = eta0
    x1 = 150
    x2 = 300
    y1 = 0
    y2 = 0
    inside = 1
    outside = 0
  [../]
[]

[AuxKernels]
  [./bnds_aux]
    type = BndsCalcAux
    variable = bnds
  [../]
[]

[Modules]
  [./PhaseField]
    [./GrandPotential]
      switching_function_names = 'hb hm'

      chemical_potentials = 'w'
      anisotropic = 'false'
      mobilities = 'chiD'
      susceptibilities = 'chi'
      free_energies_w = 'rhob rhom'

      gamma_gr = gamma
      mobility_name_gr = L
      kappa_gr = kappa
      free_energies_gr = 'omegab omegam'

      additional_ops = 'phi'
      gamma_grxop = gamma
      mobility_name_op = L_phi
      kappa_op = kappa
      free_energies_op = 'omegab omegam'
    [../]
  [../]
[]

[Materials]
  #REFERENCES
  [./constants]
    type = GenericConstantMaterial
    prop_names =  'Va      cb_eq cm_eq kb   km  mu  gamma L      L_phi  kappa  kB'
    prop_values = '0.04092 1.0   1e-5  1400 140 1.5 1.5   5.3e+3 2.3e+4 295.85 8.6173324e-5'
  [../]
  #SWITCHING FUNCTIONS
  [./switchb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'phi eta0'
    phase_etas = 'phi'
  [../]
  [./switchm]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hm
    all_etas = 'phi eta0'
    phase_etas = 'eta0'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    property_name = omegab
    coupled_variables = 'w phi'
    material_property_names = 'Va kb cb_eq'
    expression = '-0.5*w^2/Va^2/kb - w/Va*cb_eq'
    derivative_order = 2
  [../]
  [./omegam]
    type = DerivativeParsedMaterial
    property_name = omegam
    coupled_variables = 'w eta0'
    material_property_names = 'Va km cm_eq'
    expression = '-0.5*w^2/Va^2/km - w/Va*cm_eq'
    derivative_order = 2
  [../]
  [./chi]
    type = DerivativeParsedMaterial
    property_name = chi
    coupled_variables = 'w'
    material_property_names = 'Va hb hm kb km'
    expression = '(hm/km + hb/kb)/Va^2'
    derivative_order = 2
  [../]
  #DENSITIES/CONCENTRATION
  [./rhob]
    type = DerivativeParsedMaterial
    property_name = rhob
    coupled_variables = 'w'
    material_property_names = 'Va kb cb_eq'
    expression = 'w/Va^2/kb + cb_eq/Va'
    derivative_order = 1
  [../]
  [./rhom]
    type = DerivativeParsedMaterial
    property_name = rhom
    coupled_variables = 'w eta0'
    material_property_names = 'Va km cm_eq(eta0)'
    expression = 'w/Va^2/km + cm_eq/Va'
    derivative_order = 1
  [../]
  [./concentration]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'rhom hm rhob hb Va'
    expression = 'Va*(hm*rhom + hb*rhob)'
    outputs = exodus
  [../]
  [./mobility]
    type = DerivativeParsedMaterial
    material_property_names = 'chi kB'
    constant_names = 'T Em D0'
    constant_expressions = '1400 2.4 1.25e2'
    property_name = chiD
    expression = 'chi*D0*exp(-Em/kB/T)'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -sub_pc_type -pc_asm_overlap -ksp_gmres_restart -sub_ksp_type'
  petsc_options_value = ' asm      lu           1               31                 preonly'
  nl_max_its = 20
  l_max_its = 30
  l_tol = 1e-4
  nl_rel_tol = 1e-7
  nl_abs_tol = 1e-7
  start_time = 0
  dt = 2e-5
  num_steps = 3
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/nucleation/refine.i)

#
# Example derived from cahn_hilliard.i demonstrating the use of Adaptivity
# with the DiscreteNucleation system. The DiscreteNucleationMarker triggers
# mesh refinement for the nucleus geometry. It is up to the user to specify
# refinement for the physics. In this example this is done using a GradientJumpIndicator
# with a ValueThresholdMarker. The nucleation system marker and the physics marker
# must be combined using a ComboMarker to combine their effect.
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 500
  ymax = 500
  elem_type = QUAD
[]

[Modules]
  [./PhaseField]
    [./Conserved]
      [./c]
        free_energy = F
        mobility = M
        kappa = kappa_c
        solve_type = REVERSE_SPLIT
      [../]
    [../]
  [../]
[]

[ICs]
  [./c_IC]
    type = ConstantIC
    variable = c
    value = 0.2
  [../]
[]

[Materials]
  [./pfmobility]
    type = GenericConstantMaterial
    prop_names  = 'M kappa_c'
    prop_values = '1 25'
  [../]
  [./chemical_free_energy]
    # simple double well free energy
    type = DerivativeParsedMaterial
    property_name = Fc
    coupled_variables = 'c'
    constant_names       = 'barr_height  cv_eq'
    constant_expressions = '0.1          0'
    expression = 16*barr_height*c^2*(1-c)^2 # +0.01*(c*plog(c,0.005)+(1-c)*plog(1-c,0.005))
    derivative_order = 2
    outputs = exodus
  [../]
  [./probability]
    # This is a made up toy nucleation rate it should be replaced by
    # classical nucleation theory in a real simulation.
    type = ParsedMaterial
    property_name = P
    coupled_variables = c
    expression = 'if(c<0.21,c*1e-8,0)'
    outputs = exodus
  [../]
  [./nucleation]
    # The nucleation material is configured to insert nuclei into the free energy
    # tht force the concentration to go to 0.95, and holds this enforcement for 500
    # time units.
    type = DiscreteNucleation
    property_name = Fn
    op_names  = c
    op_values = 0.90
    penalty = 5
    penalty_mode = MIN
    map = map
    outputs = exodus
  [../]
  [./free_energy]
    # add the chemical and nucleation free energy contributions together
    type = DerivativeSumMaterial
    derivative_order = 2
    coupled_variables = c
    sum_materials = 'Fc Fn'
  [../]
[]

[UserObjects]
  [./inserter]
    # The inserter runs at the end of each time step to add nucleation events
    # that happened during the timestep (if it converged) to the list of nuclei
    type = DiscreteNucleationInserter
    hold_time = 50
    probability = P
    radius = 10
  [../]
  [./map]
    # The map UO runs at the beginning of a timestep and generates a per-element/qp
    # map of nucleus locations. The map is only regenerated if the mesh changed or
    # the list of nuclei was modified.
    # The map converts the nucleation points into finite area objects with a given radius.
    type = DiscreteNucleationMap
    periodic = c
    inserter = inserter
  [../]
[]

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

[BCs]
  [./Periodic]
    [./all]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Postprocessors]
  [./dt]
    type = TimestepSize
  [../]
  [./ndof]
    type = NumDOFs
  [../]
  [./rate]
    type = DiscreteNucleationData
    value = RATE
    inserter = inserter
  [../]
  [./dtnuc]
    type = DiscreteNucleationTimeStep
    inserter = inserter
    p2nucleus = 0.0005
    dt_max = 10
  [../]
  [./update]
    type = DiscreteNucleationData
    value = UPDATE
    inserter = inserter
  [../]
  [./count]
    type = DiscreteNucleationData
    value = COUNT
    inserter = inserter
  [../]
[]

[Adaptivity]
  [./Indicators]
    [./jump]
      type = GradientJumpIndicator
      variable = c
    [../]
  [../]
  [./Markers]
    [./nuc]
      type = DiscreteNucleationMarker
      map = map
    [../]
    [./grad]
      type = ValueThresholdMarker
      variable = jump
      coarsen = 0.1
      refine = 0.2
    [../]
    [./combo]
      type = ComboMarker
      markers = 'nuc grad'
    [../]
  [../]
  marker = combo
  cycles_per_step = 3
  recompute_markers_during_cycles = true
  max_h_level = 3
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu          '

  nl_max_its = 20
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  num_steps = 120

  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    growth_factor = 1.5
    cutback_factor = 0.5
    optimal_iterations = 8
    iteration_window = 2
    timestep_limiting_postprocessor = dtnuc
  [../]
[]

[Outputs]
  exodus = true
  csv = true
  print_linear_residuals = false
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_linear_fracture_energy.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l * 3 / 4'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./elastic]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'fracture_energy'
    barrier_energy = 'barrier'
    decomposition_type = strain_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./fracture_energy]
    type = DerivativeParsedMaterial
    property_name = fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = '3 * gc_prop / (8 * l) * c'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
  [./barrier_energy]
    type = ParsedMaterial
    property_name = barrier
    material_property_names = 'gc_prop l'
    expression = '3 * gc_prop / 16 / l'
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 20

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/SinteringDilute.i)

#input file to test the GrandPotentialSinteringMaterial using the dilute energy profile

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 17
  ny = 10
  xmin = 0
  xmax = 660
  ymin = 0
  ymax = 380
[]

[GlobalParams]
  op_num = 2
  var_name_base = gr
  int_width = 40
[]

[Variables]
  [./w]
    [./InitialCondition]
      type = FunctionIC
      variable = w
      function = f_w
    [../]
  [../]
  [./phi]
  [../]
  [./PolycrystalVariables]
  [../]
[]

[AuxVariables]
  [./T]
    order = CONSTANT
    family = MONOMIAL
    [./InitialCondition]
      type = FunctionIC
      variable = T
      function = f_T
    [../]
  [../]
[]

[ICs]
  [./phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '190 470'
    y_positions = '190 190'
    z_positions = '  0   0'
    radii = '150 150'
    invalue = 0
    outvalue = 1
  [../]
  [./gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 190
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 470
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
[]

[Functions]
  [./f_T]
    type = ConstantFunction
    value = 1600
  [../]
  [./f_w]
    type = ParsedFunction
    expression = '1.515e-7 * x'
  [../]
[]

[Materials]
  # Free energy coefficients for parabolic curve
  [./kv]
    type = ParsedMaterial
    property_name = kv
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.025 1571.6'
    expression = 'a*T + b'
  [../]
  # Diffusivity and mobilities
  [./chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 19.7
    c = phi
    T = T
    D0 = 2.0e11
    GBmob0 = 1.4759e9
    Q = 2.77
    Em = 2.40
    bulkindex = 1
    gbindex = 20
    surfindex = 100
  [../]
  # Equilibrium vacancy concentration
  [./cs_eq]
    type = DerivativeParsedMaterial
    property_name = cs_eq
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef Egb kB'
    constant_expressions = '2.69 2.1 8.617343e-5'
    expression = 'bnds:=gr0^2 + gr1^2; cb:=exp(-Ef/kB/T); cgb:=exp(-(Ef-Egb)/kB/T);
                cb + 4.0*(cgb-cb)*(1.0 - bnds)^2'
  [../]
  # Everything else
  [./sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 19.7
    grainboundary_energy = 9.86
    void_energy_coefficient = kv
    equilibrium_vacancy_concentration = cs_eq
    solid_energy_model = DILUTE
    outputs = exodus
  [../]

  # Concentration is only meant for output
  [./c]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'hs rhos hv rhov'
    constant_names = 'Va'
    constant_expressions = '0.04092'
    expression = 'Va*(hs*rhos + hv*rhov)'
    outputs = exodus
  [../]
[]

[Kernels]
  [./dt_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./dt_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./dt_phi]
    type = TimeDerivative
    variable = phi
  [../]
  [./dt_w]
    type = TimeDerivative
    variable = w
  [../]
[]

[AuxKernels]
  [./T_aux]
    type = FunctionAux
    variable = T
    function = f_T
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = JFNK
  dt = 1
  num_steps = 2
  nl_abs_tol = 1e-10
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/anisotropic_interfaces/GrandPotentialTwophaseAnisotropy.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmin = -4
  xmax = 4
  ymin = -4
  ymax = 4
  uniform_refine = 2
[]

[GlobalParams]
  radius = 0.5
  int_width = 0.3
  x1 = 0
  y1 = 0
  derivative_order = 2
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
[]

[AuxKernels]
  [./bnds]
    type = BndsCalcAux
    variable = bnds
    v = 'etaa0 etab0'
  [../]
[]

[ICs]
  [./w]
    type = SmoothCircleIC
    variable = w
    # note w = A*(c-cleq), A = 1.0, cleq = 0.0 ,i.e., w = c (in the matrix/liquid phase)
    outvalue = -4.0
    invalue = 0.0
  [../]
  [./etaa0]
    type = SmoothCircleIC
    variable = etaa0
    #Solid phase
    outvalue = 0.0
    invalue = 1.0
  [../]
  [./etab0]
    type = SmoothCircleIC
    variable = etab0
    #Liquid phase
    outvalue = 1.0
    invalue = 0.0
  [../]
[]

[BCs]
  [./Periodic]
    [./w]
      variable = w
      auto_direction = 'x y'
    [../]
    [./etaa0]
      variable = etaa0
      auto_direction = 'x y'
    [../]
    [./etab0]
      variable = etab0
      auto_direction = 'x y'
    [../]
  [../]
[]

[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0'
    gamma_names = 'gab'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etab0 w'
  [../]
  [./ACa0_int1]
    type = ACInterface2DMultiPhase1
    variable = etaa0
    etas = 'etab0'
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
  [../]
  [./ACa0_int2]
    type = ACInterface2DMultiPhase2
    variable = etaa0
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0'
    gamma_names = 'gab'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etaa0 w'
  [../]
  [./ACb0_int1]
    type = ACInterface2DMultiPhase1
    variable = etab0
    etas = 'etaa0'
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
  [../]
  [./ACb0_int2]
    type = ACInterface2DMultiPhase2
    variable = etab0
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
[]

[Materials]
  [./ha]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0'
    phase_etas = 'etaa0'
  [../]
  [./hb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0'
    phase_etas = 'etab0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegab
    material_property_names = 'Vm kb cbeq'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq'
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
  [../]
  [./kappaa]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
    etaa = etaa0
    etab = etab0
    outputs = exodus
    output_properties = 'kappaa'
  [../]
  [./kappab]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
    etaa = etab0
    etab = etaa0
    outputs = exodus
    output_properties = 'kappab'
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'L   D    chi  Vm   ka    caeq kb    cbeq  gab mu'
    prop_values = '1.0 1.0  0.1  1.0  10.0  0.1  10.0  0.9   4.5 10.0'
  [../]
  [./Mobility]
    type = ParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart'
  petsc_options_value = 'hypre    boomeramg      31'
  l_tol = 1.0e-3
  l_max_its = 30
  nl_max_its = 15
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1e-8
  end_time = 10.0
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.0005
    cutback_factor = 0.7
    growth_factor = 1.2
  [../]
[]

[Adaptivity]
 initial_steps = 5
 max_h_level = 3
 initial_marker = err_eta
 marker = err_bnds
[./Markers]
   [./err_eta]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_eta
   [../]
   [./err_bnds]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_bnds
   [../]
 [../]
 [./Indicators]
   [./ind_eta]
     type = GradientJumpIndicator
     variable = etaa0
    [../]
    [./ind_bnds]
      type = GradientJumpIndicator
      variable = bnds
   [../]
 [../]
[]

[Outputs]
  time_step_interval = 10
  exodus = true
[]

(modules/optimization/test/tests/optimizationreporter/constant_heat_source/adjoint_nonLinear.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 2
  ymax = 2
[]

[Variables]
  [adjoint_T]
  []
[]

[AuxVariables]
  [T]
  []
[]

[Kernels]
  [heat_conduction]
    type = MatDiffusion
    variable = adjoint_T
    diffusivity = thermal_conductivity
  []
[]

[DiracKernels]
  [pt]
    type = ReporterPointSource
    variable = adjoint_T
    x_coord_name = misfit/measurement_xcoord
    y_coord_name = misfit/measurement_ycoord
    z_coord_name = misfit/measurement_zcoord
    value_name = misfit/misfit_values
  []
[]

[Reporters]
  [misfit]
    type = OptimizationData
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'q'
    real_vector_values = '0' # Dummy value
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = adjoint_T
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = adjoint_T
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = adjoint_T
    boundary = bottom
    value = 0
  []
  [top]
    type = DirichletBC
    variable = adjoint_T
    boundary = top
    value = 0
  []
[]

[Materials]
  [steel]
    type = ParsedMaterial
    f_name = 'thermal_conductivity'
    function = '.01*T'
    args = 'T'
  []
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-8
  petsc_options_iname = '-ksp_type -pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'preonly lu       superlu_dist'
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/q'
  []
[]

[VectorPostprocessors]
  [gradient_vpp]
    type = ElementOptimizationSourceFunctionInnerProduct
    variable = adjoint_T
    function = volumetric_heat_func
  []
[]

[Outputs]
  console = false
  file_base = 'adjoint_nl'
[]

(modules/phase_field/examples/nucleation/cahn_hilliard.i)

#
# Test the DiscreteNucleation material in a toy system. The global
# concentration is above the solubility limit, but below the spinodal.
# Without further intervention no nucleation will occur in a phase
# field model. The DiscreteNucleation material will locally modify the
# free energy to coerce nuclei to grow.
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 120
  ny = 120
  xmax = 500
  ymax = 500
  elem_type = QUAD
[]

[Modules]
  [./PhaseField]
    [./Conserved]
      [./c]
        free_energy = F
        mobility = M
        kappa = kappa_c
        solve_type = REVERSE_SPLIT
      [../]
    [../]
  [../]
[]

[ICs]
  [./c_IC]
    type = RandomIC
    variable = c
    min = 0.2
    max = 0.21
  [../]
[]

[Materials]
  [./pfmobility]
    type = GenericConstantMaterial
    prop_names  = 'M kappa_c'
    prop_values = '1 25'
  [../]
  [./chemical_free_energy]
    # simple double well free energy
    type = DerivativeParsedMaterial
    property_name = Fc
    coupled_variables = 'c'
    constant_names       = 'barr_height  cv_eq'
    constant_expressions = '0.1          0'
    expression = 16*barr_height*c^2*(1-c)^2 # +0.01*(c*plog(c,0.005)+(1-c)*plog(1-c,0.005))
    derivative_order = 2
    outputs = exodus
  [../]
  [./probability]
    # This is a made up toy nucleation rate it should be replaced by
    # classical nucleation theory in a real simulation.
    type = ParsedMaterial
    property_name = P
    coupled_variables = c
    expression = c*1e-7
    outputs = exodus
  [../]
  [./nucleation]
    # The nucleation material is configured to insert nuclei into the free energy
    # tht force the concentration to go to 0.95, and holds this enforcement for 500
    # time units.
    type = DiscreteNucleation
    property_name = Fn
    op_names  = c
    op_values = 0.90
    penalty = 5
    penalty_mode = MIN
    map = map
    outputs = exodus
  [../]
  [./free_energy]
    # add the chemical and nucleation free energy contributions together
    type = DerivativeSumMaterial
    derivative_order = 2
    coupled_variables = c
    sum_materials = 'Fc Fn'
  [../]
[]

[UserObjects]
  [./inserter]
    # The inserter runs at the end of each time step to add nucleation events
    # that happend during the timestep (if it converged) to the list of nuclei
    type = DiscreteNucleationInserter
    hold_time = 100
    probability = P
    radius = 10
  [../]
  [./map]
    # The map UO runs at the beginning of a timestep and generates a per-element/qp
    # map of nucleus locations. The map is only regenerated if the mesh changed or
    # the list of nuclei was modified.
    # The map converts the nucleation points into finite area objects with a given radius.
    type = DiscreteNucleationMap
    periodic = c
    inserter = inserter
  [../]
[]

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

[BCs]
  [./Periodic]
    [./all]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Postprocessors]
  [./dt]
    type = TimestepSize
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu          '

  nl_max_its = 20
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  num_steps = 1200

  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    growth_factor = 1.5
    cutback_factor = 0.5
    optimal_iterations = 5
  [../]
[]

[Outputs]
  exodus = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_aniso_hist_false.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 20
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./All]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'strain_yy stress_yy'
        planar_formulation = PLANE_STRAIN
      [../]
    [../]
  [../]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.05 1e-6'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '127.0 70.8 70.8 127.0 70.8 127.0 73.55 73.55 73.55'
    fill_method = symmetric9
    euler_angle_1 = 30
    euler_angle_2 = 0
    euler_angle_3 = 0
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = stress_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '1.0e-6'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./av_stress_yy]
    type = ElementAverageValue
    variable = stress_yy
  [../]
  [./av_strain_yy]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solving_package'
  petsc_options_value = 'lu superlu_dist'

  nl_rel_tol = 1e-8
  l_tol = 1e-4
  l_max_its = 100
  nl_max_its = 10

  dt = 2e-6
  num_steps = 5
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/electrochem_sintering/ElectrochemicalSintering_test.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 800
  xmin = 0
  xmax = 80
[]

[GlobalParams]
  op_num = 2
  var_name_base = gr
  int_width = 4
[]

[Variables]
  [wvy]
  []
  [wvo]
  []
  [phi]
  []
  [PolycrystalVariables]
  []
  [V]
  []
[]

[AuxVariables]
  [bnds]
  []
  [negative_V]
  []
  [E_x]
    order = CONSTANT
    family = MONOMIAL
  []
  [E_y]
    order = CONSTANT
    family = MONOMIAL
  []
  [ns_cat_aux]
    order = CONSTANT
    family = MONOMIAL
  []
  [ns_an_aux]
    order = CONSTANT
    family = MONOMIAL
  []
  [T]
  []
[]

[Functions]
  [ic_func_gr0]
    type = ParsedFunction
    expression = '0.5*(1.0-tanh((x)/sqrt(2.0*2.0)))'
  []
  [ic_func_gr1]
    type = ParsedFunction
    expression = '0.5*(1.0+tanh((x)/sqrt(2.0*2.0)))'
  []
[]

[ICs]
  [gr0_IC]
    type = FunctionIC
    variable = gr0
    function = ic_func_gr0
  []
  [gr1_IC]
    type = FunctionIC
    variable = gr1
    function = ic_func_gr1
  []
  [wvy_IC]
    type = ConstantIC
    variable = wvy
    value = 2.7827
  []
  [wvo_IC]
    type = ConstantIC
    variable = wvo
    value = 2.7827
  []
  [T_IC]
    type = ConstantIC
    variable = T
    value = 1600
  []
[]

[BCs]
  [v_left]
    type = DirichletBC
    preset = true
    variable = V
    boundary = left
    value = 1e-2
  []
  [v_right]
    type = DirichletBC
    preset = true
    variable = V
    boundary = right
    value = 0
  []
  [gr0_left]
    type = DirichletBC
    preset = true
    variable = gr0
    boundary = left
    value = 0.5 #Grain boundary at left hand side of domain
  []
  [gr1_left]
    type = DirichletBC
    preset = true
    variable = gr1
    boundary = left
    value = 0.5 #Grain boundary at left hand side of domain
  []
  [wvo_right]
    type = DirichletBC
    preset = true
    variable = wvo
    boundary = right
    value = 2.7827
  []
  [wvy_right]
    type = DirichletBC
    preset = true
    variable = wvy
    boundary = right
    value = 2.7827
  []
[]

[Materials]
  # Free energy coefficients for parabolic curves
  [ks_cat]
    type = ParsedMaterial
    property_name = ks_cat
    coupled_variables = 'T'
    constant_names = 'a b Va'
    constant_expressions = '-0.0017 140.44 0.03726'
    expression = '(a*T + b) * Va^2'
  []
  [ks_an]
    type = ParsedMaterial
    property_name = ks_an
    coupled_variables = 'T'
    constant_names = 'a b Va'
    constant_expressions = '-0.0017 140.44 0.03726'
    expression = '(a*T + b) * Va^2'
  []
  [kv_cat]
    type = ParsedMaterial
    property_name = kv_cat
    material_property_names = 'ks_cat'
    expression = '10*ks_cat'
  []
  [kv_an]
    type = ParsedMaterial
    property_name = kv_an
    material_property_names = 'ks_cat'
    expression = '10*ks_cat'
  []
  # Diffusivity and mobilities
  [chiDy]
    type = GrandPotentialTensorMaterial
    f_name = chiDy
    diffusivity_name = Dvy
    solid_mobility = L
    void_mobility = Lv
    chi = chi_cat
    surface_energy = 6.24
    c = phi
    T = T
    D0 = 5.9e11
    GBmob0 = 1.60e12
    Q = 4.14
    Em = 4.25
    bulkindex = 1
    gbindex = 1
    surfindex = 1
  []
  [chiDo]
    type = GrandPotentialTensorMaterial
    f_name = chiDo
    diffusivity_name = Dvo
    solid_mobility = Lo
    void_mobility = Lvo
    chi = chi_an
    surface_energy = 6.24
    c = phi
    T = T
    D0 = 5.9e11
    GBmob0 = 1.60e12
    Q = 4.14
    Em = 4.25
    bulkindex = 1
    gbindex = 1
    surfindex = 1
  []
  # Everything else
  [ns_y_min]
    type = DerivativeParsedMaterial
    property_name = ns_y_min
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef_B Ef_GB   kB          Va_Y'
    constant_expressions = '4.37 4.37    8.617343e-5 0.03726'
    derivative_order = 2
    expression = 'bnds:=gr0^2 + gr1^2; Ef:=Ef_B + 4.0 * (Ef_GB - Ef_B) * (1.0 - bnds)^2;
              '
               '  exp(-Ef/kB/T) / Va_Y'
  []
  [ns_o_min]
    type = DerivativeParsedMaterial
    property_name = ns_o_min
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef_B Ef_GB  kB          Va_O'
    constant_expressions = '4.37 4.37   8.617343e-5 0.02484'
    derivative_order = 2
    expression = 'bnds:=gr0^2 + gr1^2; Ef:=Ef_B + 4.0 * (Ef_GB - Ef_B) * (1.0 - bnds)^2;
              '
               '  exp(-Ef/kB/T) / Va_O'
  []
  [sintering]
    type = ElectrochemicalSinteringMaterial
    chemical_potentials = 'wvy wvo'
    electric_potential = V
    void_op = phi
    Temperature = T
    surface_energy = 6.24
    grainboundary_energy = 5.18
    solid_energy_coefficients = 'kv_cat kv_cat'
    void_energy_coefficients = 'kv_cat kv_an'
    min_vacancy_concentrations_solid = 'ns_y_min ns_o_min'
    min_vacancy_concentrations_void = '26.837 40.256'
    defect_charges = '-3 2'
    solid_relative_permittivity = 30
    solid_energy_model = DILUTE
  []
  [density_chi_y]
    type = ElectrochemicalDefectMaterial
    chemical_potential = wvy
    void_op = phi
    Temperature = T
    electric_potential = V
    void_density_name = nv_cat
    solid_density_name = ns_cat
    chi_name = chi_cat
    void_energy_coefficient = kv_cat
    min_vacancy_concentration_solid = ns_y_min
    min_vacancy_concentration_void = 26.837
    solid_energy_model = DILUTE
    defect_charge = -3
    solid_relative_permittivity = 30
  []
  [density_chi_o]
    type = ElectrochemicalDefectMaterial
    chemical_potential = wvo
    void_op = phi
    Temperature = T
    electric_potential = V
    void_density_name = nv_an
    solid_density_name = ns_an
    chi_name = chi_an
    void_energy_coefficient = kv_an
    min_vacancy_concentration_solid = ns_o_min
    min_vacancy_concentration_void = 40.256
    solid_energy_model = DILUTE
    defect_charge = 2
    solid_relative_permittivity = 30
  []
  [permittivity]
    type = DerivativeParsedMaterial
    property_name = permittivity
    coupled_variables = 'phi'
    material_property_names = 'hs hv'
    constant_names = 'eps_rel_solid   eps_void_over_e'
    constant_expressions = '30              5.52e-2' #eps_void_over_e in 1/V/nm
    derivative_order = 2
    expression = '-hs * eps_rel_solid * eps_void_over_e - hv * eps_void_over_e'
  []
  [void_pre]
    type = DerivativeParsedMaterial
    property_name = void_pre
    material_property_names = 'hv'
    constant_names = 'Z_cat   Z_an nv_y_min nv_o_min'
    constant_expressions = '-3      2    26.837   40.256'
    derivative_order = 2
    expression = '-hv * (Z_cat * nv_y_min + Z_an * nv_o_min)'
  []
  [cat_mu_pre]
    type = DerivativeParsedMaterial
    property_name = cat_mu_pre
    material_property_names = 'hv kv_cat'
    constant_names = 'Z_cat'
    constant_expressions = '-3'
    derivative_order = 2
    expression = '-hv * Z_cat / kv_cat'
  []
  [an_mu_pre]
    type = DerivativeParsedMaterial
    property_name = an_mu_pre
    material_property_names = 'hv kv_an'
    constant_names = 'Z_an'
    constant_expressions = '2'
    derivative_order = 2
    expression = '-hv * Z_an / kv_an'
  []
  [cat_V_pre]
    type = DerivativeParsedMaterial
    property_name = cat_V_pre
    material_property_names = 'hv kv_cat'
    constant_names = 'Z_cat   v_scale e '
    constant_expressions = '-3      1       1'
    derivative_order = 2
    expression = 'hv * Z_cat^2 * e * v_scale / kv_cat'
  []
  [an_V_pre]
    type = DerivativeParsedMaterial
    property_name = an_V_pre
    material_property_names = 'hv kv_an'
    constant_names = 'Z_an    v_scale e '
    constant_expressions = '2       1       1'
    derivative_order = 2
    expression = 'hv * Z_an^2 * e * v_scale / kv_an'
  []
[]

#This action adds most kernels needed for grand potential model
[Modules]
  [PhaseField]
    [GrandPotential]
      switching_function_names = 'hv hs'
      anisotropic = 'true true'

      chemical_potentials = 'wvy wvo'
      mobilities = 'chiDy chiDo'
      susceptibilities = 'chi_cat chi_an'
      free_energies_w = 'nv_cat ns_cat nv_an ns_an'

      gamma_gr = gamma
      mobility_name_gr = L
      kappa_gr = kappa
      free_energies_gr = 'omegav omegas'

      additional_ops = 'phi'
      gamma_grxop = gamma
      mobility_name_op = Lv
      kappa_op = kappa
      free_energies_op = 'omegav omegas'
    []
  []
[]

[Kernels]
  [barrier_phi]
    type = ACBarrierFunction
    variable = phi
    v = 'gr0 gr1'
    gamma = gamma
    mob_name = Lv
  []
  [kappa_phi]
    type = ACKappaFunction
    variable = phi
    mob_name = Lv
    kappa_name = kappa
  []
  [Laplace]
    type = MatDiffusion
    variable = V
    diffusivity = permittivity
    args = 'phi'
  []
  [potential_void_constants]
    type = MaskedBodyForce
    variable = V
    coupled_variables = 'phi'
    mask = void_pre
  []
  [potential_cat_mu]
    type = MatReaction
    variable = V
    v = wvy
    mob_name = cat_mu_pre
  []
  [potential_an_mu]
    type = MatReaction
    variable = V
    v = wvo
    mob_name = an_mu_pre
  []
  [potential_cat_V]
    type = MatReaction
    variable = V
    mob_name = cat_V_pre
  []
  [potential_an_V]
    type = MatReaction
    variable = V
    mob_name = an_V_pre
  []
  [potential_solid_cat]
    type = MaskedExponential
    variable = V
    w = wvy
    T = T
    coupled_variables = 'phi gr0 gr1'
    mask = hs
    species_charge = -3
    n_eq = ns_y_min
  []
  [potential_solid_an]
    type = MaskedExponential
    variable = V
    w = wvo
    T = T
    coupled_variables = 'phi gr0 gr1'
    mask = hs
    species_charge = 2
    n_eq = ns_o_min
  []
[]

[AuxKernels]
  [bnds_aux]
    type = BndsCalcAux
    variable = bnds
    execute_on = 'initial timestep_end'
  []
  [negative_V]
    type = ParsedAux
    variable = negative_V
    coupled_variables = V
    expression = '-V'
  []
  [E_x]
    type = VariableGradientComponent
    variable = E_x
    gradient_variable = negative_V
    component = x
  []
  [E_y]
    type = VariableGradientComponent
    variable = E_y
    gradient_variable = negative_V
    component = y
  []
  [ns_cat_aux]
    type = MaterialRealAux
    variable = ns_cat_aux
    property = ns_cat
  []
  [ns_an_aux]
    type = MaterialRealAux
    variable = ns_an_aux
    property = ns_an
  []
[]

[Postprocessors]
  [ns_cat_total]
    type = ElementIntegralMaterialProperty
    mat_prop = ns_cat
  []
  [ns_an_total]
    type = ElementIntegralMaterialProperty
    mat_prop = ns_an
  []
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -sub_pc_type -pc_asm_overlap -ksp_gmres_restart -sub_ksp_type'
  petsc_options_value = ' asm      lu           1               31                 preonly'
  nl_max_its = 40
  l_max_its = 30
  l_tol = 1e-4
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-13
  start_time = 0
  num_steps = 2
  automatic_scaling = true
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1
    optimal_iterations = 8
    iteration_window = 2
  []
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/multiphase/DerivativeMultiPhaseMaterial.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 40
  ny = 40
  nz = 0
  xmin = -12
  xmax = 12
  ymin = -12
  ymax = 12
  elem_type = QUAD4
[]

[GlobalParams]
  # let's output all material properties for demonstration purposes
  outputs = exodus

  # prefactor on the penalty function kernels. The higher this value is, the
  # more rigorously the constraint is enforced
  penalty = 1e3
[]

#
# These AuxVariables hold the directly calculated free energy density in the
# simulation cell. They are provided for visualization purposes.
#
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    additional_free_energy = cross_energy
  [../]

  #
  # Helper kernel to cpompute the gradient contribution from interfaces of order
  # parameters evolved using the ACMultiInterface kernel
  #
  [./cross_terms]
    type = CrossTermGradientFreeEnergy
    variable = cross_energy
    interfacial_vars = 'eta1 eta2 eta3'
    #
    # The interface coefficient matrix. This should be symmetrical!
    #
    kappa_names = 'kappa11 kappa12 kappa13
                   kappa21 kappa22 kappa23
                   kappa31 kappa32 kappa33'
  [../]
[]

[Variables]
  [./c]
    order = FIRST
    family = LAGRANGE
    #
    # We set up a smooth cradial concentrtaion gradient
    # The concentration will quickly change to adapt to the preset order
    # parameters eta1, eta2, and eta3
    #
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = 0.0
      y1 = 0.0
      radius = 5.0
      invalue = 1.0
      outvalue = 0.01
      int_width = 10.0
    [../]
  [../]

  [./eta1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      #
      # Note: this initial conditions sets up a _sharp_ interface. Ideally
      # we should start with a smooth interface with a width consistent
      # with the kappa parameter supplied for the given interface.
      #
      function = 'r:=sqrt(x^2+y^2);if(r<=4,1,0)'
    [../]
  [../]
  [./eta2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      function = 'r:=sqrt(x^2+y^2);if(r>4&r<=7,1,0)'
    [../]
  [../]
  [./eta3]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      function = 'r:=sqrt(x^2+y^2);if(r>7,1,0)'
    [../]
  [../]
[]

[Kernels]
  #
  # Cahn-Hilliard kernel for the concentration variable.
  # Note that we are not using an interfcae kernel on this variable, but rather
  # rely on the interface width enforced on the order parameters. This allows us
  # to use a direct solve using the CahnHilliard kernel _despite_ only using first
  # order elements.
  #
  [./c_res]
    type = CahnHilliard
    variable = c
    f_name = F
    coupled_variables = 'eta1 eta2 eta3'
  [../]
  [./time]
    type = TimeDerivative
    variable = c
  [../]

  #
  # Order parameter eta1
  # Each order parameter is acted on by 4 kernels:
  #  1. The stock time derivative deta_i/dt kernel
  #  2. The Allen-Cahn kernel that takes a Dervative Material for the free energy
  #  3. A gradient interface kernel that includes cross terms
  #     see http://mooseframework.org/wiki/PhysicsModules/PhaseField/DevelopingModels/MultiPhaseModels/ACMultiInterface/
  #  4. A penalty contribution that forces the interface contributions h(eta)
  #     to sum up to unity
  #
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    coupled_variables = 'eta2 eta3 c'
    mob_name = L1
    f_name = F
  [../]
  [./ACInterface1]
    type = ACMultiInterface
    variable = eta1
    etas = 'eta1 eta2 eta3'
    mob_name = L1
    kappa_names = 'kappa11 kappa12 kappa13'
  [../]
  [./penalty1]
    type = SwitchingFunctionPenalty
    variable = eta1
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
  [../]

  #
  # Order parameter eta2
  #
  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    coupled_variables = 'eta1 eta3 c'
    mob_name = L2
    f_name = F
  [../]
  [./ACInterface2]
    type = ACMultiInterface
    variable = eta2
    etas = 'eta1 eta2 eta3'
    mob_name = L2
    kappa_names = 'kappa21 kappa22 kappa23'
  [../]
  [./penalty2]
    type = SwitchingFunctionPenalty
    variable = eta2
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
  [../]

  #
  # Order parameter eta3
  #
  [./deta3dt]
    type = TimeDerivative
    variable = eta3
  [../]
  [./ACBulk3]
    type = AllenCahn
    variable = eta3
    coupled_variables = 'eta1 eta2 c'
    mob_name = L3
    f_name = F
  [../]
  [./ACInterface3]
    type = ACMultiInterface
    variable = eta3
    etas = 'eta1 eta2 eta3'
    mob_name = L3
    kappa_names = 'kappa31 kappa32 kappa33'
  [../]
  [./penalty3]
    type = SwitchingFunctionPenalty
    variable = eta3
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
  [../]
[]

[BCs]
  [./Periodic]
    [./All]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Materials]
  # here we declare some of the model parameters: the mobilities and interface
  # gradient prefactors. For this example we use arbitrary numbers. In an actual simulation
  # physical mobilities would be used, and the interface gradient prefactors would
  # be readjusted to the free energy magnitudes.
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'M   kappa_c L1 L2 L3  kappa11 kappa12 kappa13 kappa21 kappa22 kappa23 kappa31 kappa32 kappa33'
    prop_values = '0.2 0.75    1  1  1   0.75    0.75    0.75    0.75    0.75    0.75    0.75    0.75    0.75   '
  [../]

  # This material sums up the individual phase contributions. It is written to the output file
  # (see GlobalParams section above) and can be used to check the constraint enforcement.
  [./etasummat]
    type = ParsedMaterial
    property_name = etasum
    material_property_names = 'h1 h2 h3'
    expression = 'h1+h2+h3'
  [../]

  # The phase contribution factors for each material point are computed using the
  # SwitchingFunctionMaterials. Each phase with an order parameter eta contributes h(eta)
  # to the global free energy density. h is a function that switches smoothly from 0 to 1
  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]
  [./switching3]
    type = SwitchingFunctionMaterial
    function_name = h3
    eta = eta3
    h_order = SIMPLE
  [../]

  # The barrier function adds a phase transformation energy barrier. It also
  # Drives order parameters toward the [0:1] interval to avoid negative or larger than 1
  # order parameters (these are set to 0 and 1 contribution by the switching functions
  # above)
  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2 eta3'
  [../]

  # We use DerivativeParsedMaterials to specify three (very) simple free energy
  # expressions for the three phases. All necessary derivatives are built automatically.
  # In a real problem these expressions can be arbitrarily complex (or even provided
  # by custom kernels).
  [./phase_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = F1
    expression = '(c-1)^2'
    coupled_variables = 'c'
  [../]
  [./phase_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = F2
    expression = '(c-0.5)^2'
    coupled_variables = 'c'
  [../]
  [./phase_free_energy_3]
    type = DerivativeParsedMaterial
    property_name = F3
    expression = 'c^2'
    coupled_variables = 'c'
  [../]

  # The DerivativeMultiPhaseMaterial ties the phase free energies together into a global free energy.
  # http://mooseframework.org/wiki/PhysicsModules/PhaseField/DevelopingModels/MultiPhaseModels/
  [./free_energy]
    type = DerivativeMultiPhaseMaterial
    property_name = F
    # we use a constant free energy (GeneriConstantmaterial property Fx)
    fi_names = 'F1  F2  F3'
    hi_names = 'h1  h2  h3'
    etas     = 'eta1 eta2 eta3'
    coupled_variables = 'c'
    W = 1
  [../]
[]

[Postprocessors]
  # The total free energy of the simulation cell to observe the energy reduction.
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
  [../]

  # for testing we also monitor the total solute amount, which should be conserved.
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
  [../]
[]

[Preconditioning]
  # This preconditioner makes sure the Jacobian Matrix is fully populated. Our
  # kernels compute all Jacobian matrix entries.
  # This allows us to use the Newton solver below.
  [./SMP]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  scheme = 'bdf2'

  # Automatic differentiation provedes a _full_ Jacobian in this example
  # so we can safely use NEWTON for a fast solve
  solve_type = 'NEWTON'

  l_max_its = 15
  l_tol = 1.0e-6

  nl_max_its = 50
  nl_rel_tol = 1.0e-6
  nl_abs_tol = 1.0e-6

  start_time = 0.0
  end_time   = 150.0

  [./TimeStepper]
    type = SolutionTimeAdaptiveDT
    dt = 0.1
  [../]
[]

[Debug]
  # show_var_residual_norms = true
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
  [./table]
    type = CSV
    delimiter = ' '
  [../]
[]

(test/tests/materials/derivative_material_interface/postprocessors.i)

#
# Test use of postprocessor values in parsed materials
#

[Mesh]
  type = GeneratedMesh
  dim = 2
[]

[Materials]
  [pp]
    type = ParsedMaterial
    expression = 'time^2'
    postprocessor_names = time
    outputs = exodus
  []
[]

[Problem]
  solve = false
[]

[Postprocessors]
  [time]
    type = TimePostprocessor
    # make sure the PostProcessor is executed early and often enough
    # when used in the ParsedMaterial (this might have to be on every NONLINEAR
    # or even LINEAR iteration!)
    execute_on = TIMESTEP_BEGIN
  []
[]

[Executioner]
  type = Transient
  num_steps = 3
[]

[Outputs]
  exodus = true
[]

(modules/combined/examples/phase_field-mechanics/Pattern1.i)

#
# Pattern example 1
#
# Phase changes driven by a combination mechanical (elastic) and chemical
# driving forces. In this three phase system a matrix phase, an oversized and
# an undersized precipitate phase compete. The chemical free energy favors a
# phase separation into either precipitate phase. A mix of both precipitate
# emerges to balance lattice expansion and contraction.
#
# This example demonstrates the use of
# * ACMultiInterface
# * SwitchingFunctionConstraintEta and SwitchingFunctionConstraintLagrange
# * DerivativeParsedMaterial
# * ElasticEnergyMaterial
# * DerivativeMultiPhaseMaterial
# * MultiPhaseStressMaterial
# which are the components to se up a phase field model with an arbitrary number
# of phases
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 80
  ny = 80
  nz = 0
  xmin = -20
  xmax = 20
  ymin = -20
  ymax = 20
  zmin = 0
  zmax = 0
  elem_type = QUAD4
[]

[GlobalParams]
  # CahnHilliard needs the third derivatives
  derivative_order = 3
  enable_jit = true
  displacements = 'disp_x disp_y'
[]

# AuxVars to compute the free energy density for outputting
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    additional_free_energy = cross_energy
  [../]
  [./cross_terms]
    type = CrossTermGradientFreeEnergy
    variable = cross_energy
    interfacial_vars = 'eta1 eta2 eta3'
    kappa_names = 'kappa11 kappa12 kappa13
                   kappa21 kappa22 kappa23
                   kappa31 kappa32 kappa33'
  [../]
[]

[Variables]
  # Solute concentration variable
  [./c]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0
      max = 0.8
      seed = 1235
    [../]
  [../]

  # Order parameter for the Matrix
  [./eta1]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  [../]
  # Order parameters for the 2 different inclusion orientations
  [./eta2]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]
  [./eta3]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]

  # Mesh displacement
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]

  # Lagrange-multiplier
  [./lambda]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1.0
  [../]
[]

[Kernels]
  # Set up stress divergence kernels
  [./TensorMechanics]
  [../]

  # Cahn-Hilliard kernels
  [./c_res]
    type = CahnHilliard
    variable = c
    f_name = F
    args = 'eta1 eta2 eta3'
  [../]
  [./time]
    type = TimeDerivative
    variable = c
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 1
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    args = 'eta2 eta3 c'
    mob_name = L1
    f_name = F
  [../]
  [./ACInterface1]
    type = ACMultiInterface
    variable = eta1
    etas = 'eta1 eta2 eta3'
    mob_name = L1
    kappa_names = 'kappa11 kappa12 kappa13'
  [../]
  [./lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name   = h1
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 2
  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    args = 'eta1 eta3 c'
    mob_name = L2
    f_name = F
  [../]
  [./ACInterface2]
    type = ACMultiInterface
    variable = eta2
    etas = 'eta1 eta2 eta3'
    mob_name = L2
    kappa_names = 'kappa21 kappa22 kappa23'
  [../]
  [./lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name   = h2
    lambda = lambda
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 3
  [./deta3dt]
    type = TimeDerivative
    variable = eta3
  [../]
  [./ACBulk3]
    type = AllenCahn
    variable = eta3
    args = 'eta1 eta2 c'
    mob_name = L3
    f_name = F
  [../]
  [./ACInterface3]
    type = ACMultiInterface
    variable = eta3
    etas = 'eta1 eta2 eta3'
    mob_name = L3
    kappa_names = 'kappa31 kappa32 kappa33'
  [../]
  [./lagrange3]
    type = SwitchingFunctionConstraintEta
    variable = eta3
    h_name   = h3
    lambda = lambda
  [../]

  # Lagrange-multiplier constraint kernel for lambda
  [./lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    etas    = 'eta1 eta2 eta3'
    h_names = 'h1   h2   h3'
    epsilon = 1e-6
  [../]
[]

[Materials]
  # declare a few constants, such as mobilities (L,M) and interface gradient prefactors (kappa*)
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'M   kappa_c  L1 L2 L3  kappa11 kappa12 kappa13 kappa21 kappa22 kappa23 kappa31 kappa32 kappa33'
    prop_values = '0.2 0        1  1  1   2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00    2.00   '
  [../]

  # We use this to output the level of constraint enforcement
  # ideally it should be 0 everywhere, if the constraint is fully enforced
  [./etasummat]
    type = ParsedMaterial
    property_name = etasum
    coupled_variables = 'eta1 eta2 eta3'
    material_property_names = 'h1 h2 h3'
    expression = 'h1+h2+h3-1'
    outputs = exodus
  [../]

  # This parsed material creates a single property for visualization purposes.
  # It will be 0 for phase 1, -1 for phase 2, and 1 for phase 3
  [./phasemap]
    type = ParsedMaterial
    property_name = phase
    coupled_variables = 'eta2 eta3'
    expression = 'if(eta3>0.5,1,0)-if(eta2>0.5,1,0)'
    outputs = exodus
  [../]

  # matrix phase
  [./elasticity_tensor_1]
    type = ComputeElasticityTensor
    base_name = phase1
    C_ijkl = '3 3'
    fill_method = symmetric_isotropic
  [../]
  [./strain_1]
    type = ComputeSmallStrain
    base_name = phase1
    displacements = 'disp_x disp_y'
  [../]
  [./stress_1]
    type = ComputeLinearElasticStress
    base_name = phase1
  [../]

  # oversized phase
  [./elasticity_tensor_2]
    type = ComputeElasticityTensor
    base_name = phase2
    C_ijkl = '7 7'
    fill_method = symmetric_isotropic
  [../]
  [./strain_2]
    type = ComputeSmallStrain
    base_name = phase2
    displacements = 'disp_x disp_y'
    eigenstrain_names = eigenstrain
  [../]
  [./stress_2]
    type = ComputeLinearElasticStress
    base_name = phase2
  [../]
  [./eigenstrain_2]
    type = ComputeEigenstrain
    base_name = phase2
    eigen_base = '0.02'
    eigenstrain_name = eigenstrain
  [../]

  # undersized phase
  [./elasticity_tensor_3]
    type = ComputeElasticityTensor
    base_name = phase3
    C_ijkl = '7 7'
    fill_method = symmetric_isotropic
  [../]
  [./strain_3]
    type = ComputeSmallStrain
    base_name = phase3
    displacements = 'disp_x disp_y'
    eigenstrain_names = eigenstrain
  [../]
  [./stress_3]
    type = ComputeLinearElasticStress
    base_name = phase3
  [../]
  [./eigenstrain_3]
    type = ComputeEigenstrain
    base_name = phase3
    eigen_base = '-0.05'
    eigenstrain_name = eigenstrain
  [../]

  # switching functions
  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]
  [./switching3]
    type = SwitchingFunctionMaterial
    function_name = h3
    eta = eta3
    h_order = SIMPLE
  [../]

  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2 eta3'
  [../]

  # chemical free energies
  [./chemical_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = Fc1
    expression = '4*c^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = Fc2
    expression = '(c-0.9)^2-0.4'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_3]
    type = DerivativeParsedMaterial
    property_name = Fc3
    expression = '(c-0.9)^2-0.5'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # elastic free energies
  [./elastic_free_energy_1]
    type = ElasticEnergyMaterial
    base_name = phase1
    f_name = Fe1
    derivative_order = 2
    args = 'c' # should be empty
  [../]
  [./elastic_free_energy_2]
    type = ElasticEnergyMaterial
    base_name = phase2
    f_name = Fe2
    derivative_order = 2
    args = 'c' # should be empty
  [../]
  [./elastic_free_energy_3]
    type = ElasticEnergyMaterial
    base_name = phase3
    f_name = Fe3
    derivative_order = 2
    args = 'c' # should be empty
  [../]

  # phase free energies (chemical + elastic)
  [./phase_free_energy_1]
    type = DerivativeSumMaterial
    property_name = F1
    sum_materials = 'Fc1 Fe1'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./phase_free_energy_2]
    type = DerivativeSumMaterial
    property_name = F2
    sum_materials = 'Fc2 Fe2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./phase_free_energy_3]
    type = DerivativeSumMaterial
    property_name = F3
    sum_materials = 'Fc3 Fe3'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # global free energy
  [./free_energy]
    type = DerivativeMultiPhaseMaterial
    f_name = F
    fi_names = 'F1  F2  F3'
    hi_names = 'h1  h2  h3'
    etas     = 'eta1 eta2 eta3'
    coupled_variables = 'c'
    W = 3
  [../]

  # Generate the global stress from the phase stresses
  [./global_stress]
    type = MultiPhaseStressMaterial
    phase_base = 'phase1 phase2 phase3'
    h          = 'h1     h2     h3'
  [../]
[]

[BCs]
  # the boundary conditions on the displacement enforce periodicity
  # at zero total shear and constant volume
  [./bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom'
    value = 0
  [../]
  [./top_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'top'
    value = 0
  [../]
  [./left_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0
  [../]
  [./right_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'right'
    value = 0
  [../]

  [./Periodic]
    [./disp_x]
      auto_direction = 'y'
    [../]
    [./disp_y]
      auto_direction = 'x'
    [../]

    # all other phase field variables are fully periodic
    [./c]
      auto_direction = 'x y'
    [../]
    [./eta1]
      auto_direction = 'x y'
    [../]
    [./eta2]
      auto_direction = 'x y'
    [../]
    [./eta3]
      auto_direction = 'x y'
    [../]
    [./lambda]
      auto_direction = 'x y'
    [../]
  [../]
[]

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

# We monitor the total free energy and the total solute concentration (should be constant)
[Postprocessors]
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
  [../]
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      ilu'

  l_max_its = 30
  nl_max_its = 10
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  num_steps = 200

  [./TimeStepper]
    type = SolutionTimeAdaptiveDT
    dt = 0.1
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
  [./table]
    type = CSV
    delimiter = ' '
  [../]
[]

[Debug]
  # show_var_residual_norms = true
[]

(test/tests/materials/derivative_material_interface/ad_material_chaining.i)

#
# This test validates the correct application of the chain rule to coupled
# material properties within DerivativeParsedMaterials
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 5
  ny = 5
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
[]

[Variables]
  [./eta1]
  [../]
  [./eta2]
  [../]
[]

[BCs]
  [./left]
    variable = eta1
    boundary = left
    type = DirichletBC
    value = 0
  [../]
  [./right]
    variable = eta1
    boundary = right
    type = DirichletBC
    value = 1
  [../]
  [./top]
    variable = eta2
    boundary = top
    type = DirichletBC
    value = 0
  [../]
  [./bottom]
    variable = eta2
    boundary = bottom
    type = DirichletBC
    value = 1
  [../]
[]

[Materials]
  # T1 := (eta1+1)^4
  [./term]
    type = ADDerivativeParsedMaterial
    property_name= T1
    coupled_variables = 'eta1'
    expression = '(eta1+1)^4'
    derivative_order = 4
  [../]

  # in this material we substitute T1 explicitly
  [./full]
    type = ADDerivativeParsedMaterial
    coupled_variables = 'eta1 eta2'
    property_name = F1
    expression = '(1-eta2)^4+(eta1+1)^4'
  [../]
  # in this material we utilize the T1 derivative material property
  [./subs]
    type = ADDerivativeParsedMaterial
    coupled_variables = 'eta1 eta2'
    property_name = F2
    expression = '(1-eta2)^4+T1'
    material_property_names = 'T1(eta1)'
  [../]

  # calculate differences between the explicit and indirect substitution version
  # the use if the T1 property should include dT1/deta1 contributions!
  # This also demonstrated the explicit use of material property derivatives using
  # the D[...] syntax.
  [./diff0]
    type = ADParsedMaterial
    property_name = D0
    expression = '(F1-F2)^2'
    material_property_names = 'F1 F2'
  [../]
  [./diff1]
    type = ADParsedMaterial
    property_name = D1
    expression = '(dF1-dF2)^2'
    material_property_names = 'dF1:=D[F1,eta1] dF2:=D[F2,eta1]'
  [../]
  [./diff2]
    type = ADParsedMaterial
    property_name = D2
    expression = '(d2F1-d2F2)^2'
    material_property_names = 'd2F1:=D[F1,eta1,eta1] d2F2:=D[F2,eta1,eta1]'
  [../]

  # check that explicitly pulling a derivative yields the correct result by
  # taking the difference of the manually calculated 1st derivative of T1 and the
  # automatic derivative dT1 pulled in through dT1:=D[T1,eta1]
  [./diff3]
    type = ADParsedMaterial
    property_name = E0
    expression = '(dTd1-(4*(eta1+1)^3))^2'
    coupled_variables = eta1
    material_property_names = 'dTd1:=D[T1,eta1]'
  [../]
[]

[Kernels]
  [./eta1diff]
    type = Diffusion
    variable = eta1
  [../]
  [./eta2diff]
    type = Diffusion
    variable = eta2
  [../]
[]

[Postprocessors]
  [./D0]
    type = ADElementIntegralMaterialProperty
    mat_prop = D0
  [../]
  [./D1]
    type = ADElementIntegralMaterialProperty
    mat_prop = D1
  [../]
  [./D2]
    type = ADElementIntegralMaterialProperty
    mat_prop = D2
  [../]
  [./E0]
    type = ADElementIntegralMaterialProperty
    mat_prop = E0
  [../]
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  l_tol = 1e-03
[]

[Outputs]
  execute_on = 'TIMESTEP_END'
  csv = true
  print_linear_residuals = false
[]

(modules/phase_field/test/tests/grain_tracker_test/grain_tracker_reserve.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 40
  ny = 40
  xmin = 0
  xmax = 100
  ymin = 0
  ymax = 100
  elem_type = QUAD4
[]

[AuxVariables]
  [./c]
  [../]
[]

[Variables]
  [./gr0]
  [../]
  [./gr1]
  [../]
[]

[ICs]
  [./gr0]
    type = MultiSmoothCircleIC
    variable = gr0
    invalue = 1.0
    outvalue = 0.0001
    bubspac = 20.0
    numbub = 2
    radius = 10.0
    int_width = 12.0
    radius_variation = 0.2
    radius_variation_type = uniform
  [../]
  [./c_IC]
    type = SmoothCircleIC
    int_width = 12.0
    x1 = 50
    y1 = 50
    radius = 10.0
    outvalue = 0
    variable = c
    invalue = 1
  [../]
[]

[Kernels]
  [./ie_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./diff_gr0]
    type = Diffusion
    variable = gr0
  [../]
  [./ie_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./diff_gr1]
    type = Diffusion
    variable = gr1
  [../]
  [./source]
    type = MaskedBodyForce
    variable = gr1
    function = t
    mask = mask
  [../]
[]

[Materials]
  [./mask]
    type = ParsedMaterial
    expression = 'c'
    property_name = mask
    coupled_variables = 'c'
  [../]
[]

[Postprocessors]
  [./grain_tracker]
    type = GrainTracker

    # Reserve the first "op" variable
    reserve_op = 1
    threshold = 0.1
    connecting_threshold = 0.001
    variable = 'gr0 gr1'
    execute_on = 'initial timestep_end'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  num_steps = 6
  dt = 0.25
[]

[Outputs]
  exodus = true
[]

[Problem]
  kernel_coverage_check = false
[]

(modules/phase_field/examples/multiphase/GrandPotential3Phase.i)

# This is an example of implementation of the multi-phase, multi-order parameter
# grand potential based phase-field model described in Phys. Rev. E, 98, 023309
# (2018). It includes 3 phases with 1 grain of each phase. This example was used
# to generate the results shown in Fig. 3 of the paper.

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 60
  xmin = -15
  xmax = 15
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
  [./etad0]
  [../]
[]

[ICs]
  [./IC_etaa0]
    type = FunctionIC
    variable = etaa0
    function = ic_func_etaa0
  [../]
  [./IC_etab0]
    type = FunctionIC
    variable = etab0
    function = ic_func_etab0
  [../]
  [./IC_etad0]
    type = ConstantIC
    variable = etad0
    value = 0.1
  [../]
  [./IC_w]
    type = FunctionIC
    variable = w
    function = ic_func_w
  [../]
[]

[Functions]
  [./ic_func_etaa0]
    type = ParsedFunction
    expression = '0.9*0.5*(1.0-tanh((x)/sqrt(2.0)))'
  [../]
  [./ic_func_etab0]
    type = ParsedFunction
    expression = '0.9*0.5*(1.0+tanh((x)/sqrt(2.0)))'
  [../]
  [./ic_func_w]
    type = ParsedFunction
    expression = 0
  [../]
[]

[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0 etad0'
    gamma_names = 'gab   gad'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
    coupled_variables = 'etab0 etad0 w'
  [../]
  [./ACa0_int]
    type = ACInterface
    variable = etaa0
    kappa_name = kappa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0 etad0'
    gamma_names = 'gab   gbd'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
    coupled_variables = 'etaa0 etad0 w'
  [../]
  [./ACb0_int]
    type = ACInterface
    variable = etab0
    kappa_name = kappa
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
# Order parameter eta_delta0
  [./ACd0_bulk]
    type = ACGrGrMulti
    variable = etad0
    v =           'etaa0 etab0'
    gamma_names = 'gad   gbd'
  [../]
  [./ACd0_sw]
    type = ACSwitching
    variable = etad0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
    coupled_variables = 'etaa0 etab0 w'
  [../]
  [./ACd0_int]
    type = ACInterface
    variable = etad0
    kappa_name = kappa
  [../]
  [./ed0_dot]
    type = TimeDerivative
    variable = etad0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
    coupled_variables = 'etaa0 etab0 etad0'
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
    args = ''
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    coupled_variables = 'etaa0 etab0 etad0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    coupled_variables = 'etaa0 etab0 etad0'
  [../]
  [./coupled_etad0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etad0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    coupled_variables = 'etaa0 etab0 etad0'
  [../]
[]

[Materials]
  [./ha_test]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etaa0'
  [../]
  [./hb_test]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etab0'
  [../]
  [./hd_test]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hd
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etad0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
    derivative_order = 2
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegab
    material_property_names = 'Vm kb cbeq'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq'
    derivative_order = 2
  [../]
  [./omegad]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegad
    material_property_names = 'Vm kd cdeq'
    expression = '-0.5*w^2/Vm^2/kd-w/Vm*cdeq'
    derivative_order = 2
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
    derivative_order = 2
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
    derivative_order = 2
  [../]
  [./rhod]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhod
    material_property_names = 'Vm kd cdeq'
    expression = 'w/Vm^2/kd + cdeq/Vm'
    derivative_order = 2
  [../]
  [./c]
    type = ParsedMaterial
    material_property_names = 'Vm rhoa rhob rhod ha hb hd'
    expression = 'Vm * (ha * rhoa + hb * rhob + hd * rhod)'
    property_name = c
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'kappa_c  kappa   L   D    Vm   ka    caeq kb    cbeq  kd    cdeq  gab gad gbd  mu  tgrad_corr_mult'
    prop_values = '0        1       1.0 1.0  1.0  10.0  0.1  10.0  0.9   10.0  0.5   1.5 1.5 1.5  1.0 0.0'
  [../]
  [./Mobility]
    type = DerivativeParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
    derivative_order = 2
  [../]
  [./chi]
    type = DerivativeParsedMaterial
    property_name = chi
    material_property_names = 'Vm ha(etaa0,etab0,etad0) ka hb(etaa0,etab0,etad0) kb hd(etaa0,etab0,etad0) kd'
    expression = '(ha/ka + hb/kb + hd/kd) / Vm^2'
    coupled_variables = 'etaa0 etab0 etad0'
    derivative_order = 2
  [../]
[]

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

[VectorPostprocessors]
  [./etaa0]
    type = LineValueSampler
    variable = etaa0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  [../]
  [./etab0]
    type = LineValueSampler
    variable = etab0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  [../]
  [./etad0]
    type = LineValueSampler
    variable = etad0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  [../]
[]

[Executioner]
  type = Transient
  nl_max_its = 15
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = -pc_type
  petsc_options_value = asm
  l_max_its = 15
  l_tol = 1.0e-3
  nl_rel_tol = 1.0e-8
  start_time = 0.0
  num_steps = 20
  nl_abs_tol = 1e-10
  dt = 1.0
[]

[Outputs]
  [./exodus]
    type = Exodus
    execute_on = 'initial timestep_end final'
    time_step_interval = 1
  [../]
  [./csv]
    type = CSV
    execute_on = 'initial timestep_end final'
    time_step_interval = 1
  [../]
[]

(modules/combined/test/tests/optimization/invOpt_nonlinear/adjoint.i)

[Executioner]
  type = Steady
  solve_type = NEWTON
  line_search = none
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-12
  nl_max_its = 100
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Mesh]
[]

[Variables]
  [adjointT]
  []
[]

[AuxVariables]
  [forwardT]
  []
  [dDdTgradT]
    order  = CONSTANT
    family = MONOMIAL_VEC
  []
[]

[Kernels]
  [heat_conduction]
    type = ADHeatConduction
    thermal_conductivity = 'linearized_conductivity'
    variable = adjointT
  []
  [advection]
    type = LevelSetAdvection
    velocity = dDdTgradT
    variable = adjointT
  []
[]

[AuxKernels]
  [dDdTgradT]
    type = ADFunctorElementalGradientAux
    functor = forwardT
    variable = dDdTgradT
    factor_matprop = 'dDdT'
  []
[]

[Materials]
  [LinearizedConductivity]
    type = ADParsedMaterial
    f_name = 'linearized_conductivity'
    function = '10+500*forwardT'
    args = 'forwardT'
  []
  [dDdT]
    type = ADParsedMaterial
    f_name = 'dDdT'
    function = '500'
    args = 'forwardT'
  []
[]


[DiracKernels]
  [pt]
    type = ReporterPointSource
    variable = adjointT
    x_coord_name = misfit/measurement_xcoord
    y_coord_name = misfit/measurement_ycoord
    z_coord_name = misfit/measurement_zcoord
    value_name   = misfit/misfit_values
  []
[]

[Reporters]
  [misfit]
    type = OptimizationData
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'heat_source'
    real_vector_values = '0' # Dummy
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = adjointT
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = adjointT
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = adjointT
    boundary = bottom
    value = 0
  []
  [top]
    type = DirichletBC
    variable = adjointT
    boundary = top
    value = 0
  []
[]

[VectorPostprocessors]
  [gradient_vpp]
    type = ElementOptimizationSourceFunctionInnerProduct
    function = volumetric_heat_func
    variable = adjointT
  []
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/heat_source'
  []
[]

[Outputs]
  console = false
[]

(modules/optimization/test/tests/optimizationreporter/constant_heat_source/forward_nonLinear.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 2
  ymax = 2
[]

[Variables]
  [T]
    initial_condition = 100
  []
[]

[Kernels]
  [heat_conduction]
    type = MatDiffusion
    variable = T
    diffusivity = thermal_conductivity
  []
  [heat_source]
    type = BodyForce
    function = volumetric_heat_func
    variable = T
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = T
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = T
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = T
    boundary = bottom
    value = 200
  []
  [top]
    type = DirichletBC
    variable = T
    boundary = top
    value = 100
  []
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/q'
  []
[]

[Materials]
  [steel]
    type = ParsedMaterial
    f_name = 'thermal_conductivity'
    function = '.01*T'
    args = 'T'
  []
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-8
  petsc_options_iname = '-ksp_type -pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'preonly lu       superlu_dist'
[]

[Reporters]
  [measure_data]
    type = OptimizationData
    variable = T
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'q'
    real_vector_values = '0' # Dummy value
  []
[]

[Outputs]
  console = false
  file_base = 'forward_nl'
[]

(modules/solid_mechanics/test/tests/crystal_plasticity/hcp_single_crystal/hcp_volumetric_eigenstrain.i)

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 2
  ny = 2
  nz = 2
  elem_type = HEX8
[]

[AuxVariables]
  [temperature]
    order = FIRST
    family = LAGRANGE
  []
  [e_xtalpl_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_xtalpl_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [ev_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_xtalpl_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [fv_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [fv_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [fv_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_zz]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = FINITE
  incremental = true
  add_variables = true
  additional_generate_output = 'stress_zz stress_xx stress_yy vonmises_stress l2norm_strain'
  additional_material_output_order = FIRST
[]

[Functions]
  [temperature_ramp]
    type = ParsedFunction
    expression = 'if(t<=1500.0, 600.0 + t/6.0, 850.0)'
  []
[]

[AuxKernels]
  [temperature]
    type = FunctionAux
    variable = temperature
    function = 'temperature_ramp'
    execute_on = timestep_begin
  []
  [e_xtalpl_xx]
    type = RankTwoAux
    variable = e_xtalpl_xx
    rank_two_tensor = total_lagrangian_strain
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [e_xtalpl_yy]
    type = RankTwoAux
    variable = e_xtalpl_yy
    rank_two_tensor = total_lagrangian_strain
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [ev_zz]
    type = RankTwoAux
    variable = ev_zz
    rank_two_tensor = void_eigenstrain
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [e_xtalpl_zz]
    type = RankTwoAux
    variable = e_xtalpl_zz
    rank_two_tensor = total_lagrangian_strain
    index_i = 2
    index_j = 2
    execute_on = timestep_end
  []
  [fv_xx]
    type = RankTwoAux
    variable = fv_xx
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [fv_yy]
    type = RankTwoAux
    variable = fv_yy
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [fv_zz]
    type = RankTwoAux
    variable = fv_zz
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [fp_xx]
    type = RankTwoAux
    variable = fp_xx
    rank_two_tensor = plastic_deformation_gradient
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [fp_yy]
    type = RankTwoAux
    variable = fp_yy
    rank_two_tensor = plastic_deformation_gradient
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [fp_zz]
    type = RankTwoAux
    variable = fp_zz
    rank_two_tensor = plastic_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [f_xx]
    type = RankTwoAux
    variable = f_xx
    rank_two_tensor = deformation_gradient
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [f_yy]
    type = RankTwoAux
    variable = f_yy
    rank_two_tensor = deformation_gradient
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [f_zz]
    type = RankTwoAux
    variable = f_zz
    rank_two_tensor = deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
[]

[BCs]
  [symmy]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
  [symmx]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [symmz]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0
  []
  [tdisp]
    type = DirichletBC
    variable = disp_z
    boundary = front
    value = 0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeElasticityTensorCP
    C_ijkl = '1.684e5 1.214e5 1.214e5 1.684e5 1.214e5 1.684e5 0.754e5 0.754e5 0.754e5'
    fill_method = symmetric9
  []
  [stress]
    type = ComputeMultipleCrystalPlasticityStress
    crystal_plasticity_models = 'trial_xtalpl'
    eigenstrain_names = void_eigenstrain
    tan_mod_type = exact
    line_search_method = CUT_HALF
    use_line_search = true
    maximum_substep_iteration = 10
  []
  [trial_xtalpl]
    type = CrystalPlasticityHCPDislocationSlipBeyerleinUpdate
    number_slip_systems = 15
    slip_sys_file_name = hcp_aprismatic_capyramidal_slip_sys.txt
    unit_cell_dimension = '2.934e-7 2.934e-7 4.657e-7' #Ti, in mm, https://materialsproject.org/materials/mp-46/
    temperature = temperature
    initial_forest_dislocation_density = 15.0e3
    initial_substructure_density = 1.0e3
    slip_system_modes = 2
    number_slip_systems_per_mode = '3 12'
    lattice_friction_per_mode = '9 22' #Knezevic et al MSEA 654 (2013)
    effective_shear_modulus_per_mode = '4.7e2 4.7e2' #Ti, in MPa, https://materialsproject.org/materials/mp-46/
    burgers_vector_per_mode = '2.934e-7 6.586e-7' #Ti, in mm, https://materialsproject.org/materials/mp-46/
    slip_generation_coefficient_per_mode = '1.25e5 2.25e7' #from Beyerlein and Tome 2008 IJP
    normalized_slip_activiation_energy_per_mode = '3.73e-3 3.2e-2' #from Beyerlein and Tome 2008 IJP
    slip_energy_proportionality_factor_per_mode = '330 100' #from Beyerlein and Tome 2008 IJP
    substructure_rate_coefficient_per_mode = '355 0.4' #from Capolungo et al MSEA (2009)
    applied_strain_rate = 0.001
    gamma_o = 1.0e-3
    Hall_Petch_like_constant_per_mode = '0.2 0.2' #Estimated to match graph in Capolungo et al MSEA (2009), Figure 2
    grain_size = 20.0e-3 #20 microns, Beyerlein and Tome IJP (2008)
  []
  [void_eigenstrain]
    type = ComputeCrystalPlasticityVolumetricEigenstrain
    eigenstrain_name = void_eigenstrain
    deformation_gradient_name = volumetric_deformation_gradient
    mean_spherical_void_radius = void_radius
    spherical_void_number_density = void_density
  []
  [void_density]
    type = ParsedMaterial
    property_name = void_density
    coupled_variables = temperature
    expression = 'if(temperature<611.0, 0.0,
                    if(temperature<=835.0, 2.387e13 *(temperature - 611.0) / 1344.0, 0.0))' #1/mm^3, gives an eigenstrain of 1.0e-5 with radius=1.0e-6mm
    # outputs = exodus
  []
  [void_radius]
    type = GenericConstantMaterial
    prop_names = void_radius
    prop_values = '1.0e-6'  ##1 nm avg particle radius
  []
[]

[Postprocessors]
  [stress_zz]
    type = ElementAverageValue
    variable = stress_zz
  []
  [e_xtalpl_xx]
    type = ElementAverageValue
    variable = e_xtalpl_xx
  []
  [e_xtalpl_yy]
    type = ElementAverageValue
    variable = e_xtalpl_yy
  []
  [ev_zz]
    type = ElementAverageValue
    variable = ev_zz
  []
  [e_xtalpl_zz]
    type = ElementAverageValue
    variable = e_xtalpl_zz
[]
  [fv_xx]
    type = ElementAverageValue
    variable = fv_xx
  []
  [fv_yy]
    type = ElementAverageValue
    variable = fv_yy
  []
  [fv_zz]
    type = ElementAverageValue
    variable = fv_zz
  []
  [temperature]
    type = ElementAverageValue
    variable = temperature
  []
  [fp_xx]
    type = ElementAverageValue
    variable = fp_xx
  []
  [fp_yy]
    type = ElementAverageValue
    variable = fp_yy
  []
  [fp_zz]
    type = ElementAverageValue
    variable = fp_zz
  []
  [f_xx]
    type = ElementAverageValue
    variable = f_xx
  []
  [f_yy]
    type = ElementAverageValue
    variable = f_yy
  []
  [f_zz]
    type = ElementAverageValue
    variable = f_zz
  []
[]

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

[Debug]
  show_var_residual_norms = true
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_asm_overlap -sub_pc_type -ksp_type -ksp_gmres_restart'
  petsc_options_value = ' asm      2              lu            gmres     200'
  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-10
  nl_abs_step_tol = 1e-10

  dt = 10.0
  dtmin = 1e-4
  # end_time = 10
  num_steps = 10
[]

[Outputs]
  csv = true
  [console]
    type = Console
    # max_rows = 5
  []
[]

(modules/heat_transfer/test/tests/transient_heat/transient_heat_derivatives.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [temp]
    order = FIRST
    family = LAGRANGE
    initial_condition = 2
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []

  [ie]
    type = HeatConductionTimeDerivative
    variable = temp
    specific_heat_dT = specific_heat_dT
    density_name_dT = density_dT
  []
[]

[Functions]
  [spheat]
    type = ParsedFunction
    expression = 't^4'
  []
  [thcond]
    type = ParsedFunction
    expression = 'exp(t)'
  []
[]

[BCs]
  [bottom]
    type = DirichletBC
    variable = temp
    boundary = 1
    value = 4
  []

  [top]
    type = DirichletBC
    variable = temp
    boundary = 2
    value = 1
  []
[]

[Materials]
  [constant]
    type = HeatConductionMaterial
    thermal_conductivity_temperature_function = thcond
    specific_heat_temperature_function = spheat
    temp = temp
  []
  [density]
    type = ParsedMaterial
    property_name = density
    coupled_variables = temp
    expression = 'temp^3 + 2/temp'
  []
  [density_dT]
    type = ParsedMaterial
    property_name = density_dT
    coupled_variables = temp
    expression = '3 * temp^2 - 2/temp/temp'
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  num_steps = 1
  dt = .1
  nl_max_its = 10
  dtmin = .1
[]

[Postprocessors]
  [avg]
    type = ElementAverageValue
    variable = temp
  []
[]

[Outputs]
  csv = true
[]

(modules/stochastic_tools/examples/surrogates/combined/trans_diff_2d/trans_diff_sub.i)

[Functions]
  [src_func]
    type = ParsedFunction
    expression = "1000*sin(f*t)"
    symbol_names = 'f'
    symbol_values = '20'
  []
[]

[Mesh]
  [msh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 100
    xmin = -0.5
    xmax = 0.5
    ny = 100
    ymin = -0.5
    ymax = 0.5
  []
  [source_domain]
    type = ParsedSubdomainMeshGenerator
    input = msh
    combinatorial_geometry = '(x<0.1 & x>-0.1) & (y<0.1 & y>-0.1)'
    block_id=1
  []
[]

[Variables]
  [T]
    initial_condition = 300
  []
[]

[Kernels]
  [diffusion]
    type = MatDiffusion
    variable = T
    diffusivity = diff_coeff
  []
  [source]
    type = BodyForce
    variable = T
    function = src_func
    block = 1
  []
  [time_deriv]
    type = TimeDerivative
    variable = T
  []
[]

[Materials]
  [diff_coeff]
    type = ParsedMaterial
    property_name = diff_coeff
    coupled_variables = 'T'
    constant_names = 'C'
    constant_expressions = 0.02
    expression = 'C * pow(300/T, 2)'
  []
[]

[BCs]
  [neumann_all]
    type = NeumannBC
    variable = T
    boundary = 'left right top bottom'
    value = 0
  []
[]

[Executioner]
  type = Transient
  num_steps = 100
  dt = 0.01
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  nl_rel_tol = 1e-6
  l_abs_tol = 1e-6
  timestep_tolerance = 1e-6
[]

[Postprocessors]
  [max]
    type = NodalExtremeValue
    variable = T
  []
  [min]
    type = NodalExtremeValue
    variable = T
    value_type = min
  []
  [time_max]
    type = TimeExtremeValue
    postprocessor = max
  []
  [time_min]
    type = TimeExtremeValue
    postprocessor = min
    value_type = min
  []
[]

(modules/optimization/test/tests/reporter/TransientSyntheticDataCreator/transient_synthetic_data.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 16
    ny = 16
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
  []
[]

[Variables/u]
[]

[Functions]
  [rxn_func]
    type = ParsedFunction
    expression = 'exp(x * y) - 1'
  []
[]

[Materials]
  [ad_dc_prop]
    type = ADParsedMaterial
    expression = '1 + u'
    coupled_variables = 'u'
    property_name = dc_prop
  []
  [ad_rxn_prop]
    type = ADGenericFunctionMaterial
    prop_values = 'rxn_func'
    prop_names = rxn_prop
  []
  [ad_neg_rxn_prop]
    type = ADParsedMaterial
    expression = '-rxn_prop'
    material_property_names = 'rxn_prop'
    property_name = 'neg_rxn_prop'
  []
[]

[Kernels]
  [udot]
    type = ADTimeDerivative
    variable = u
  []
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = dc_prop
  []
  [reaction]
    type = ADMatReaction
    variable = u
    reaction_rate = neg_rxn_prop
  []
  [src]
    type = ADBodyForce
    variable = u
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u
    boundary = 'left bottom'
    value = 0
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  dt = 0.1
  end_time = 1
[]

[Reporters]
  [sample]
    type = TransientSyntheticDataCreator
    variable = u
    single_set_of_measurement_points = '
    0.75   0.75   0
    0.9375 0.9375 0
    1      0.9375 0
    1      1      0'

    measurement_times_for_all_points = '0.1 0.2 0.3'
  []
[]

[AuxVariables]
  [reaction_rate]
  []
[]

[AuxKernels]
  [reaction_rate_aux]
    type = FunctionAux
    variable = reaction_rate
    function = rxn_func
    execute_on = TIMESTEP_END
  []
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = 'FINAL'
  []
[]

(test/tests/materials/derivative_material_interface/parsed_material.i)

#
# Test the parsed function free enery Allen-Cahn Bulk kernel
#

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmin = 0
  xmax = 1
[]

[AuxVariables]
  [./eta]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      function = x
    [../]
  [../]
[]

[Materials]
  [./consts]
    type = ParsedMaterial
    coupled_variables  = 'eta'
    expression ='(eta-0.5)^2'
    outputs = exodus
  [../]
[]

[Problem]
  solve = false
[]

[Executioner]
  type = Steady
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/electromagnetics/test/tests/benchmarks/waveguide2D/waveguide2D_test.i)

# Test for EMRobinBC in port and absorbing modes with simple electric plane wave
# 2D, vacuum-filled waveguide with conducting walls
# u^2 + k^2*u = 0, 0 < x < 80, 0 < y < 10, u: R -> C
# k = 2*pi*freq/c, freq = 20e6 Hz, c = 3e8 m/s

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

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

[Functions]
  [inc_y]
    type = ParsedFunction
    expression = 'sin(pi * y / 10)'
  []
[]

[Kernels]
  [diffusion_real]
    type = Diffusion
    variable = E_real
  []
  [coeffField_real]
    type = ADMatReaction
    reaction_rate = kSquared
    variable = E_real
  []
  [diffusion_imaginary]
    type = Diffusion
    variable = E_imag
  []
  [coeffField_imaginary]
    type = ADMatReaction
    reaction_rate = kSquared
    variable = E_imag
  []
[]

[BCs]
  [top_real]
    type = DirichletBC
    value = 0
    variable = E_real
    boundary = top
  []
  [bottom_real]
    type = DirichletBC
    value = 0
    variable = E_real
    boundary = bottom
  []
  [port_real]
    type = EMRobinBC
    coeff_real = -0.27706242940220277  # -sqrt(k^2 - (pi/10)^2)
    sign = positive
    profile_func_real = inc_y
    profile_func_imag = 0
    field_real = E_real
    field_imaginary = E_imag
    variable = E_real
    component = real
    mode = port
    boundary = port
  []
  [exit_real]
    type = EMRobinBC
    coeff_real = 0.27706242940220277
    sign = negative
    field_real = E_real
    field_imaginary = E_imag
    variable = E_real
    component = real
    mode = absorbing
    boundary = exit
  []
  [top_imaginary]
    type = DirichletBC
    value = 0
    variable = E_imag
    boundary = top
  []
  [bottom_imaginary]
    type = DirichletBC
    value = 0
    variable = E_imag
    boundary = bottom
  []
  [port_imaginary]
    type = EMRobinBC
    coeff_real = -0.27706242940220277
    sign = positive
    profile_func_real = inc_y
    profile_func_imag = 0
    field_real = E_real
    field_imaginary = E_imag
    variable = E_imag
    component = imaginary
    mode = port
    boundary = port
  []
  [exit_imaginary]
    type = EMRobinBC
    coeff_real = 0.27706242940220277
    sign = negative
    field_real = E_real
    field_imaginary = E_imag
    variable = E_imag
    component = imaginary
    mode = absorbing
    boundary = exit
  []
[]

[Materials]
  [kSquared]
    type = ADParsedMaterial
    property_name = kSquared
    expression = '0.4188790204786391^2'
  []
[]

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

[Outputs]
  exodus = true
  print_linear_residuals = true
[]

(modules/navier_stokes/test/tests/finite_element/ins/cg-dg-hybrid/mms/lid-driven-skewed/hybrid-skewed-vortex.i)

rho=1
mu=1

[Mesh]
  [gen_mesh]
    type = FileMeshGenerator
    file = skewed.msh
  []
  [corner_node]
    type = ExtraNodesetGenerator
    new_boundary = 'pinned_node'
    nodes = '0'
    input = gen_mesh
  []
[]

[Variables]
  [u]
    family = MONOMIAL
    order = SECOND
  []
  [v]
    family = MONOMIAL
    order = SECOND
  []
  [pressure][]
[]

[Kernels]
  [momentum_x_convection]
    type = ADConservativeAdvection
    variable = u
    velocity = 'velocity'
  []
  [momentum_x_diffusion]
    type = Diffusion
    variable = u
  []
  [momentum_x_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = u
    pressure = pressure
    component = 0
  []
  [u_forcing]
    type = BodyForce
    variable = u
    function = forcing_u
  []
  [momentum_y_convection]
    type = ADConservativeAdvection
    variable = v
    velocity = 'velocity'
  []
  [momentum_y_diffusion]
    type = Diffusion
    variable = v
  []
  [momentum_y_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = v
    pressure = pressure
    component = 1
  []
  [v_forcing]
    type = BodyForce
    variable = v
    function = forcing_v
  []
  [mass]
    type = ADConservativeAdvection
    variable = pressure
    velocity = velocity
    advected_quantity = -1
  []
[]

[DGKernels]
  [momentum_x_convection]
    type = ADDGAdvection
    variable = u
    velocity = 'velocity'
  []
  [momentum_x_diffusion]
    type = DGDiffusion
    variable = u
    sigma = 6
    epsilon = -1
  []
  [momentum_y_convection]
    type = ADDGAdvection
    variable = v
    velocity = 'velocity'
  []
  [momentum_y_diffusion]
    type = DGDiffusion
    variable = v
    sigma = 6
    epsilon = -1
  []
[]

[BCs]
  [u_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = u
    sigma = 6
    epsilon = -1
    function = exact_u
  []
  [v_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = v
    sigma = 6
    epsilon = -1
    function = exact_v
  []
  [pressure_pin]
    type = FunctionDirichletBC
    variable = pressure
    boundary = 'pinned_node'
    function = 'exact_p'
  []
[]

[Materials]
  [rho]
    type = ADGenericConstantMaterial
    prop_names = 'rho'
    prop_values = '${rho}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = u
    v = v
  []
  [rhou]
    type = ADParsedMaterial
    property_name = 'rhou'
    coupled_variables = 'u'
    material_property_names = 'rho'
    expression = 'rho*u'
  []
  [rhov]
    type = ADParsedMaterial
    property_name = 'rhov'
    coupled_variables = 'v'
    material_property_names = 'rho'
    expression = 'rho*v'
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'x^2*(1-x)^2*(2*y-6*y^2+4*y^3)'
  []
  [exact_v]
    type = ParsedFunction
    expression = '-y^2*(1-y)^2*(2*x-6*x^2+4*x^3)'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'x*(1-x)-2/12'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '-4*mu/rho*(-1+2*y)*(y^2-6*x*y^2+6*x^2*y^2-y+6*x*y-6*x^2*y+3*x^2-6*x^3+3*x^4)+1-2*x+4*x^3'
            '*y^2*(2*y^2-2*y+1)*(y-1)^2*(-1+2*x)*(x-1)^3'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '4*mu/rho*(-1+2*x)*(x^2-6*y*x^2+6*x^2*y^2-x+6*x*y-6*x*y^2+3*y^2-6*y^3+3*y^4)+4*y^3*x^2*(2'
            '*x^2-2*x+1)*(x-1)^2*(-1+2*y)*(y-1)^3'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       NONZERO               mumps'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2u]
    type = ElementL2Error
    variable = u
    function = exact_u
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2v]
    variable = v
    function = exact_v
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2p]
    variable = pressure
    function = exact_p
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/materials/derivative_material_interface/material_chaining.i)

#
# This test validates the correct application of the chain rule to coupled
# material properties within DerivativeParsedMaterials
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 5
  ny = 5
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
[]

[Variables]
  [./eta1]
  [../]
  [./eta2]
  [../]
[]

[BCs]
  [./left]
    variable = eta1
    boundary = left
    type = DirichletBC
    value = 0
  [../]
  [./right]
    variable = eta1
    boundary = right
    type = DirichletBC
    value = 1
  [../]
  [./top]
    variable = eta2
    boundary = top
    type = DirichletBC
    value = 0
  [../]
  [./bottom]
    variable = eta2
    boundary = bottom
    type = DirichletBC
    value = 1
  [../]
[]

[Materials]
  # T1 := (eta1+1)^4
  [./term]
    type = DerivativeParsedMaterial
    property_name= T1
    coupled_variables = 'eta1'
    expression = '(eta1+1)^4'
    derivative_order = 4
  [../]

  # in this material we substitute T1 explicitly
  [./full]
    type = DerivativeParsedMaterial
    coupled_variables = 'eta1 eta2'
    property_name = F1
    expression = '(1-eta2)^4+(eta1+1)^4'
  [../]
  # in this material we utilize the T1 derivative material property
  [./subs]
    type = DerivativeParsedMaterial
    coupled_variables = 'eta1 eta2'
    property_name = F2
    expression = '(1-eta2)^4+T1'
    material_property_names = 'T1(eta1)'
  [../]

  # calculate differences between the explicit and indirect substitution version
  # the use if the T1 property should include dT1/deta1 contributions!
  # This also demonstrated the explicit use of material property derivatives using
  # the D[...] syntax.
  [./diff0]
    type = ParsedMaterial
    property_name = D0
    expression = '(F1-F2)^2'
    material_property_names = 'F1 F2'
  [../]
  [./diff1]
    type = ParsedMaterial
    property_name = D1
    expression = '(dF1-dF2)^2'
    material_property_names = 'dF1:=D[F1,eta1] dF2:=D[F2,eta1]'
  [../]
  [./diff2]
    type = ParsedMaterial
    property_name = D2
    expression = '(d2F1-d2F2)^2'
    material_property_names = 'd2F1:=D[F1,eta1,eta1] d2F2:=D[F2,eta1,eta1]'
  [../]

  # check that explicitly pulling a derivative yields the correct result by
  # taking the difference of the manually calculated 1st derivative of T1 and the
  # automatic derivative dT1 pulled in through dT1:=D[T1,eta1]
  [./diff3]
    type = ParsedMaterial
    property_name = E0
    expression = '(dTd1-(4*(eta1+1)^3))^2'
    coupled_variables = eta1
    material_property_names = 'dTd1:=D[T1,eta1]'
  [../]
[]

[Kernels]
  [./eta1diff]
    type = Diffusion
    variable = eta1
  [../]
  [./eta2diff]
    type = Diffusion
    variable = eta2
  [../]
[]

[Postprocessors]
  [./D0]
    type = ElementIntegralMaterialProperty
    mat_prop = D0
  [../]
  [./D1]
    type = ElementIntegralMaterialProperty
    mat_prop = D1
  [../]
  [./D2]
    type = ElementIntegralMaterialProperty
    mat_prop = D2
  [../]
  [./E0]
    type = ElementIntegralMaterialProperty
    mat_prop = E0
  [../]
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  l_tol = 1e-03
[]

[Outputs]
  execute_on = 'TIMESTEP_END'
  csv = true
  print_linear_residuals = false
[]

(modules/navier_stokes/test/tests/finite_element/ins/cg-dg-hybrid/channel-flow/channel-hybrid.i)

mu = 1.1
rho = 1.1

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 10
    ymin = -1
    ymax = 1
    nx = 100
    ny = 20
  []
[]

[Variables]
  [u]
    family = MONOMIAL
  []
  [v]
    family = MONOMIAL
  []
  [pressure][]
[]

[Kernels]
  [momentum_x_convection]
    type = ADConservativeAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = MatDiffusion
    variable = u
    diffusivity = 'mu'
  []
  [momentum_x_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = u
    pressure = pressure
    component = 0
  []
  [momentum_y_convection]
    type = ADConservativeAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = MatDiffusion
    variable = v
    diffusivity = 'mu'
  []
  [momentum_y_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = v
    pressure = pressure
    component = 1
  []
  [mass]
    type = ADConservativeAdvection
    variable = pressure
    velocity = velocity
    advected_quantity = -1
  []
[]

[DGKernels]
  [momentum_x_convection]
    type = ADDGAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = DGDiffusion
    variable = u
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
  [momentum_y_convection]
    type = ADDGAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = DGDiffusion
    variable = v
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
[]

[Functions]
  [v_inlet]
    type = ParsedVectorFunction
    expression_x = '1'
  []
[]

[BCs]
  [u_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'bottom top'
    variable = u
    sigma = 6
    epsilon = -1
    function = '0'
    diff = 'mu'
  []
  [v_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'bottom top'
    variable = v
    sigma = 6
    epsilon = -1
    function = '0'
    diff = 'mu'
  []
  [u_in]
    type = ADConservativeAdvectionBC
    boundary = 'left'
    variable = u
    velocity_function = v_inlet
    primal_dirichlet_value = 1
    primal_coefficient = 'rho'
  []
  [v_in]
    type = ADConservativeAdvectionBC
    boundary = 'left'
    variable = v
    velocity_function = v_inlet
    primal_dirichlet_value = 0
    primal_coefficient = 'rho'
  []
  [p_in]
    type = ADConservativeAdvectionBC
    boundary = 'left'
    variable = pressure
    velocity_function = v_inlet
    advected_quantity = -1
  []
  [u_out]
    type = ADConservativeAdvectionBC
    boundary = 'right'
    variable = u
    velocity_mat_prop = 'velocity'
    advected_quantity = 'rhou'
  []
  [v_out]
    type = ADConservativeAdvectionBC
    boundary = 'right'
    variable = v
    velocity_mat_prop = 'velocity'
    advected_quantity = 'rhov'
  []
  [p_out]
    type = DirichletBC
    variable = pressure
    boundary = 'right'
    value = 0
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho'
    prop_values = '${rho}'
  []
  [const_reg]
    type = GenericConstantMaterial
    prop_names = 'mu'
    prop_values = '${mu}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = u
    v = v
  []
  [rhou]
    type = ADParsedMaterial
    property_name = 'rhou'
    coupled_variables = 'u'
    material_property_names = 'rho'
    expression = 'rho*u'
  []
  [rhov]
    type = ADParsedMaterial
    property_name = 'rhov'
    coupled_variables = 'v'
    material_property_names = 'rho'
    expression = 'rho*v'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu       NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(modules/optimization/examples/diffusion_reaction/forward_and_adjoint.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 16
    ny = 16
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
  []
[]

[Variables/u]
[]

[Reporters]
  [params]
    type = ConstantReporter
    real_vector_names = 'reaction_rate'
    real_vector_values = '0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0' # Dummy
    outputs = none
  []
  [data]
    type = OptimizationData
    variable = u
    measurement_file = forward_exact_csv_sample_0011.csv
    file_xcoord = measurement_xcoord
    file_ycoord = measurement_ycoord
    file_zcoord = measurement_zcoord
    file_time = measurement_time
    file_value = simulation_values
    outputs = none
  []
[]

[Functions]
  [rxn_func]
    type = ParameterMeshFunction
    exodus_mesh = parameter_mesh_out.e
    parameter_name = params/reaction_rate
  []
[]

[Materials]
  [ad_dc_prop]
    type = ADParsedMaterial
    expression = '1 + u'
    coupled_variables = 'u'
    property_name = dc_prop
  []
  [ad_rxn_prop]
    type = ADGenericFunctionMaterial
    prop_values = 'rxn_func'
    prop_names = rxn_prop
  []
  #ADMatReaction includes a negative sign in residual evaluation, so we need to
  #reverse this with a negative reaction rate. However, we wanted the parameter
  #to remain positive, which is why there is one object to evaluate function
  #and another to flip it's sign for the kernel
  [ad_neg_rxn_prop]
    type = ADParsedMaterial
    expression = '-rxn_prop'
    material_property_names = 'rxn_prop'
    property_name = 'neg_rxn_prop'
  []
[]

[Kernels]
  [udot]
    type = ADTimeDerivative
    variable = u
  []
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = dc_prop
  []
  [reaction]
    type = ADMatReaction
    variable = u
    reaction_rate = neg_rxn_prop
  []
  [src]
    type = ADBodyForce
    variable = u
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u
    boundary = 'left bottom'
    value = 0
  []
[]

[Executioner]
  type = TransientAndAdjoint
  forward_system = nl0
  adjoint_system = adjoint
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  dt = 0.1
  end_time = 1

  nl_rel_tol = 1e-12
[]

[Problem]
  nl_sys_names = 'nl0 adjoint'
  kernel_coverage_check = false
  skip_nl_system_check = true
[]

[Variables]
  [u_adjoint]
    initial_condition = 0
    solver_sys = adjoint
    outputs = none
  []
[]

[DiracKernels]
  [misfit]
    type = ReporterTimePointSource
    variable = u_adjoint
    value_name = data/misfit_values
    x_coord_name = data/measurement_xcoord
    y_coord_name = data/measurement_ycoord
    z_coord_name = data/measurement_zcoord
    time_name = data/measurement_time
  []
[]

[VectorPostprocessors]
  [adjoint]
    type = ElementOptimizationReactionFunctionInnerProduct
    variable = u_adjoint
    forward_variable = u
    function = rxn_func
    execute_on = ADJOINT_TIMESTEP_END
    outputs = none
  []
[]

[AuxVariables]
  [reaction_rate]
  []
[]

[AuxKernels]
  [reaction_rate_aux]
    type = FunctionAux
    variable = reaction_rate
    function = rxn_func
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [u1]
    type = PointValue
    variable = u
    point = '0.25 0.25 0'
  []
  [u2]
    type = PointValue
    variable = u
    point = '0.75 0.75 0'
  []
  [u3]
    type = PointValue
    variable = u
    point = '1 1 0'
  []
[]

[Outputs]
  exodus = true
  console = false
  csv = true
[]

(modules/combined/test/tests/optimization/compliance_sensitivity/three_materials_thermal.i)

vol_frac = 0.4
cost_frac = 0.4

power = 4

# Stiffness (not optimized in this test)
E0 = 1.0e-6
E1 = 0.2
E2 = 0.6
E3 = 1.0

# Densities
rho0 = 1.0e-6
rho1 = 0.4
rho2 = 0.7
rho3 = 1.0

# Costs
C0 = 1.0e-6
C1 = 0.5
C2 = 0.8
C3 = 1.0

# Thermal conductivity
TC0 = 1.0e-6
TC1 = 0.2
TC2 = 0.6
TC3 = 1.0

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 40
    xmin = 0
    xmax = 40
    ymin = 0
    ymax = 40
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push_left]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push_left
    coord = '20 0 0'
  []
  [push_center]
    type = ExtraNodesetGenerator
    input = push_left
    new_boundary = push_center
    coord = '40 0 0'
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 100.0
  []
[]

[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Tc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cost]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]

# [ICs]
#   [mat_den]
#     type = RandomIC
#     seed = 4
#     variable = mat_den
#     max = '${fparse vol_frac+0.25}'
#     min = '${fparse vol_frac-0.25}'
#   []
# []

[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [top]
    type = DirichletBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = DirichletBC
    variable = temp
    boundary = left
    value = 0
  []
[]

[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-6 # -3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-6 # -3
    mass = 1
  []
[]

[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${TC0}-${TC1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${TC0}-A1*${rho0}^${power}; TC1:=A1*mat_den^${power}+B1; "
                 "A2:=(${TC1}-${TC2})/(${rho1}^${power}-${rho2}^${power}); "
                 "B2:=${TC1}-A2*${rho1}^${power}; TC2:=A2*mat_den^${power}+B2; "
                 "A3:=(${TC2}-${TC3})/(${rho2}^${power}-${rho3}^${power}); "
                 "B3:=${TC2}-A3*${rho2}^${power}; TC3:=A3*mat_den^${power}+B3; "
                 "if(mat_den<${rho1},TC1,if(mat_den<${rho2},TC2,TC3))"
    coupled_variables = 'mat_den'
    property_name = thermal_cond
    outputs = 'exodus'
  []
  [thermal_compliance]
    type = ThermalCompliance
    temperature = temp
    thermal_conductivity = thermal_cond
    outputs = 'exodus'
  []
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${E0}-${E1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${E0}-A1*${rho0}^${power}; E1:=A1*mat_den^${power}+B1; "
                 "A2:=(${E1}-${E2})/(${rho1}^${power}-${rho2}^${power}); "
                 "B2:=${E1}-A2*${rho1}^${power}; E2:=A2*mat_den^${power}+B2; "
                 "A3:=(${E2}-${E3})/(${rho2}^${power}-${rho3}^${power}); "
                 "B3:=${E2}-A3*${rho2}^${power}; E3:=A3*mat_den^${power}+B3; "
                 "if(mat_den<${rho1},E1,if(mat_den<${rho2},E2,E3))"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^(1/${power})-${rho1}^(1/${power})); "
                 "B1:=${C0}-A1*${rho0}^(1/${power}); C1:=A1*mat_den^(1/${power})+B1; "
                 "A2:=(${C1}-${C2})/(${rho1}^(1/${power})-${rho2}^(1/${power})); "
                 "B2:=${C1}-A2*${rho1}^(1/${power}); C2:=A2*mat_den^(1/${power})+B2; "
                 "A3:=(${C2}-${C3})/(${rho2}^(1/${power})-${rho3}^(1/${power})); "
                 "B3:=${C2}-A3*${rho2}^(1/${power}); C3:=A3*mat_den^(1/${power})+B3; "
                 "if(mat_den<${rho1},C1,if(mat_den<${rho2},C2,C3))"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
  []
  [CostDensity]
    type = ParsedMaterial
    property_name = CostDensity
    coupled_variables = 'mat_den Cost'
    expression = 'mat_den*Cost'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = Cost_mat
    outputs = 'exodus'
  []
  [tc]
    type = ThermalSensitivity
    design_density = mat_den
    thermal_conductivity = thermal_cond
    temperature = temp
    outputs = 'exodus'
  []
[]

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

[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 4
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 4
    weights = linear
    prop_name = cost_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_thermal]
    type = RadialAverage
    radius = 4
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdateTwoConstraints
    density_sensitivity = Dc
    cost_density_sensitivity = Cc
    cost = Cost
    cost_fraction = ${cost_frac}
    design_density = mat_den
    volume_fraction = ${vol_frac}

    bisection_lower_bound = 0
    bisection_upper_bound = 1.0e12 # 100

    use_thermal_compliance = true
    thermal_sensitivity = Tc
    # Only account for thermal optimizxation
    weight_mechanical_thermal = '0 1'

    relative_tolerance = 1.0e-8
    bisection_move = 0.05
    adaptive_move = false
    execute_on = TIMESTEP_BEGIN
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Cc
  [calc_sense_cost]
    type = SensitivityFilter
    density_sensitivity = Cc
    design_density = mat_den
    filter_UO = rad_avg_cost
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Tc
  [calc_sense_thermal]
    type = SensitivityFilter
    density_sensitivity = Tc
    design_density = mat_den
    filter_UO = rad_avg_thermal
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-10
  dt = 1.0
  num_steps = 12
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [right_flux]
    type = SideDiffusiveFluxAverage
    variable = temp
    boundary = right
    diffusivity = 10
  []
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralMaterialProperty
    mat_prop = CostDensity
  []
  [cost_frac]
    type = ParsedPostprocessor
    expression = 'cost / mesh_volume'
    pp_names = 'cost mesh_volume'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [objective_thermal]
    type = ElementIntegralMaterialProperty
    mat_prop = thermal_compliance
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

(test/tests/userobjects/nodal_patch_recovery/nodal_patch_recovery.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[UserObjects]
  [u_patch]
    type = NodalPatchRecoveryMaterialProperty
    patch_polynomial_order = FIRST
    property = 'u'
    execute_on = 'TIMESTEP_END'
  []
[]

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

[Kernels]
  [dt]
    type = TimeDerivative
    variable = v
  []
  [diff]
    type = Diffusion
    variable = v
  []
[]

[AuxVariables]
  [u_recovered]
  []
  [u_nodal]
  []
  [u_diff]
  []
[]

[AuxKernels]
  [u_recovered]
    type = NodalPatchRecoveryAux
    variable = u_recovered
    nodal_patch_recovery_uo = u_patch
    execute_on = 'TIMESTEP_END'
  []
  [u_nodal]
    type = ParsedAux
    variable = u_nodal
    expression = v^2
    coupled_variables = v
  []
  [u_diff]
    type = ParsedAux
    variable = u_diff
    expression = u_nodal-u_recovered
    coupled_variables = 'u_nodal u_recovered'
  []
[]

[BCs]
  [fix_left]
    type = FunctionDirichletBC
    variable = v
    boundary = 'left'
    function = y+1
  []
  [fix_right]
    type = DirichletBC
    variable = v
    boundary = 'right'
    value = 0
  []
[]

[Materials]
  [u]
    type = ParsedMaterial
    expression = v^2
    property_name = u
    coupled_variables = v
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON

  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'

  automatic_scaling = true

  dt = 0.4
  num_steps = 5

  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-10
[]

[Outputs]
  exodus = true
  print_linear_residuals = false
[]

(modules/solid_mechanics/test/tests/ad_viscoplasticity_stress_update/lps_single.i)

# This test provides an example of an individual LPS viscoplasticity model

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmax = 0.002
  ymax = 0.002
[]

[Physics/SolidMechanics/QuasiStatic/All]
  strain = FINITE
  add_variables = true
  generate_output = 'strain_xx strain_yy strain_xy hydrostatic_stress vonmises_stress'
  use_automatic_differentiation = true
[]

[Functions]
  [./pull]
    type = PiecewiseLinear
    x = '0 0.1'
    y = '0 1e-5'
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1e10
    poissons_ratio = 0.3
  [../]
  [./stress]
    type = ADComputeMultipleInelasticStress
    inelastic_models = lps
    outputs = all
  [../]
  [./porosity]
    type = ADPorosityFromStrain
    initial_porosity = 0.1
    inelastic_strain = 'combined_inelastic_strain'
    outputs = 'all'
  [../]

  [./lps]
    type = ADViscoplasticityStressUpdate
    coefficient = 'coef'
    power = 3
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./coef]
    type = ADParsedMaterial
    property_name = coef
    # Example of creep power law
    expression = '1e-18 * exp(-4e4 / 1.987 / 1200)'
  [../]
[]

[BCs]
  [./no_disp_x]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./no_disp_y]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./pull_disp_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = top
    function = pull
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 0.01
  end_time = 0.12
[]

[Postprocessors]
  [./disp_x]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  [../]
  [./disp_y]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
  [./avg_hydro]
    type = ElementAverageValue
    variable = hydrostatic_stress
  [../]
  [./avg_vonmises]
    type = ElementAverageValue
    variable = vonmises_stress
  [../]
  [./dt]
    type = TimestepSize
  [../]
  [./num_lin]
    type = NumLinearIterations
    outputs = console
  [../]
  [./num_nonlin]
    type = NumNonlinearIterations
    outputs = console
  [../]
  [./eff_creep_strain]
    type = ElementAverageValue
    variable = effective_viscoplasticity
  [../]
  [./porosity]
    type = ElementAverageValue
    variable = porosity
  [../]
[]

[Outputs]
  csv = true
[]

(modules/combined/test/tests/phase_field_fracture/void2d_iso.i)

[Mesh]
  type = FileMesh
  file = void2d_mesh.xda
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./All]
        add_variables = true
        strain = SMALL
        additional_generate_output = stress_yy
      [../]
    [../]
  [../]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        mobility = L
        kappa = kappa_op
      [../]
    [../]
  [../]
[]

[Functions]
  [./tfunc]
    type = ParsedFunction
    expression = t
  [../]
  [./void_prop_func]
    type = ParsedFunction
    expression = 'rad:=0.2;m:=50;r:=sqrt(x^2+y^2);1-exp(-(r/rad)^m)+1e-8'
  [../]
  [./gb_prop_func]
    type = ParsedFunction
    expression = 'rad:=0.2;thk:=0.05;m:=50;sgnx:=1-exp(-(x/rad)^m);v:=sgnx*exp(-(y/thk)^m);0.005*(1-v)+0.001*v'
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = tfunc
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'l visco'
    prop_values = '0.01 0.1'
  [../]
  [./pfgc]
    type = GenericFunctionMaterial
    prop_names = 'gc_prop'
    prop_values = 'gb_prop_func'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
    elasticity_tensor_prefactor = void_prop_func
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'fracture_energy'
    decomposition_type = strain_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./fracture_energy]
    type = DerivativeParsedMaterial
    property_name = fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK

  petsc_options_iname = '-pc_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      lu           1'

  nl_rel_tol = 1e-9
  nl_max_its = 10
  l_tol = 1e-4
  l_max_its = 40

  dt = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/solid_mechanics/test/tests/crystal_plasticity/cp_eigenstrains/volumetric_eigenstrain_increase.i)

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  [cube]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 2
    ny = 2
    nz = 2
    elem_type = HEX27
  []
[]

[AuxVariables]
  [temperature]
    order = FIRST
    family = LAGRANGE
  []
  [linear_void_strain]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [pk2_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [tau_0]
    order = FIRST
    family = MONOMIAL
  []
  [tau_10]
    order = FIRST
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = FINITE
  incremental = true
  add_variables = true
[]

[Functions]
  [temperature_ramp]
    type = ParsedFunction
    expression = '600.0 + t'
  []
[]

[AuxKernels]
  [temperature]
    type = FunctionAux
    variable = temperature
    function = 'temperature_ramp'
    execute_on = timestep_begin
  []
  [linear_void_strain]
    type = MaterialRealAux
    variable = linear_void_strain
    property = equivalent_linear_change
    execute_on = timestep_end
  []
  [e_void_xx]
    type = RankTwoAux
    variable = e_void_xx
    rank_two_tensor = void_eigenstrain
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [e_void_yy]
    type = RankTwoAux
    variable = e_void_yy
    rank_two_tensor = void_eigenstrain
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [e_void_zz]
    type = RankTwoAux
    variable = e_void_zz
    rank_two_tensor = void_eigenstrain
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [f_void_zz]
    type = RankTwoAux
    variable = f_void_zz
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [pk2_zz]
    type = RankTwoAux
    variable = pk2_zz
    rank_two_tensor = second_piola_kirchhoff_stress
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [fp_zz]
    type = RankTwoAux
    variable = fp_zz
    rank_two_tensor = plastic_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [tau_0]
    type = MaterialStdVectorAux
    variable = tau_0
    property = applied_shear_stress
    index = 0
    execute_on = timestep_end
  []
  [tau_10]
    type = MaterialStdVectorAux
    variable = tau_10
    property = applied_shear_stress
    index = 10
    execute_on = timestep_end
  []
[]

[BCs]
  [symmy]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
  [symmx]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [symmz]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0
  []
  [hold_front]
    type = DirichletBC
    variable = disp_z
    boundary = front
    value = 0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeElasticityTensorCP
    C_ijkl = '1.98e5 1.25e5 1.25e5 1.98e5 1.25e5 1.98e5 1.22e5 1.22e5 1.22e5'
    fill_method = symmetric9
  []
  [stress]
    type = ComputeMultipleCrystalPlasticityStress
    crystal_plasticity_models = 'trial_xtalpl'
    eigenstrain_names = void_eigenstrain
    tan_mod_type = exact
    line_search_method = CUT_HALF
    use_line_search = true
    maximum_substep_iteration = 5
  []

  [trial_xtalpl]
    type = CrystalPlasticityKalidindiUpdate
    number_slip_systems = 12
    slip_sys_file_name = input_slip_sys.txt
  []
  [void_eigenstrain]
    type = ComputeCrystalPlasticityVolumetricEigenstrain
    eigenstrain_name = void_eigenstrain
    deformation_gradient_name = volumetric_deformation_gradient
    mean_spherical_void_radius = void_radius
    spherical_void_number_density = void_density
  []
  [void_density]
    type = ParsedMaterial
    property_name = void_density
    coupled_variables = temperature
    expression = '1.0e8 * (temperature - 600.0)'
  []
  [void_radius]
    type = GenericConstantMaterial
    prop_names = void_radius
    prop_values = '1.0e-6'  ##1 nm avg particle radius
  []
[]

[Postprocessors]
  [linear_void_strain]
    type = ElementAverageValue
    variable = linear_void_strain
  []
  [e_void_xx]
    type = ElementAverageValue
    variable = e_void_xx
  []
  [e_void_yy]
    type = ElementAverageValue
    variable = e_void_yy
  []
  [e_void_zz]
    type = ElementAverageValue
    variable = e_void_zz
  []
  [f_void_zz]
    type = ElementAverageValue
    variable = f_void_zz
  []
  [density]
    type = ElementAverageMaterialProperty
    mat_prop = void_density
    execute_on = TIMESTEP_END
  []
  [radius]
    type = ElementAverageMaterialProperty
    mat_prop = void_radius
    execute_on = TIMESTEP_END
  []
  [pk2_zz]
   type = ElementAverageValue
   variable = pk2_zz
  []
  [fp_zz]
    type = ElementAverageValue
    variable = fp_zz
  []
  [tau_0]
    type = ElementAverageValue
    variable = tau_0
  []
  [tau_10]
    type = ElementAverageValue
    variable = tau_10
  []
[]

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

[Executioner]
  type = Transient
  solve_type = 'NEWTON'

  petsc_options = '-snes_converged_reason'
  petsc_options_iname = '-pc_type -pc_asm_overlap -sub_pc_type -ksp_type -ksp_gmres_restart'
  petsc_options_value = ' asm      2              lu            gmres     200'

  line_search = 'none'
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-8
  nl_forced_its = 1

  dt = 1.0
  dtmin = 0.1
  end_time  = 5.0
[]

[Outputs]
  csv = true
  perf_graph = true
[]

(test/tests/materials/derivative_material_interface/construction_order.i)

#
# Test the the getDefaultMaterialProperty in DerivativeMaterialInterface.
# This test should only pass, if the construction order of the Materials
# using this interface does not influence the outcome.
#

[Mesh]
  [gen]
   type = GeneratedMeshGenerator
    dim = 2
    nx = 5
    ny = 1
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 0.1
    elem_type = QUAD4
  []
[]

[GlobalParams]
  derivative_order = 2
[]

[Variables]
  [./c]
    [./InitialCondition]
      type = FunctionIC
      function = x
    [../]
  [../]
[]

[Kernels]
  [./dummy1]
    type = Diffusion
    variable = c
  [../]
  [./dummy2]
    type = TimeDerivative
    variable = c
  [../]
[]

[Materials]
  # derivatives used both before and after being declared
  [./sum_a_1]
    type = DerivativeSumMaterial
    property_name = Fa1
    sum_materials = 'Fa'
    coupled_variables = 'c'
    outputs = exodus
  [../]
  [./free_energy_a]
    type = DerivativeParsedMaterial
    property_name = Fa
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_a_2]
    type = DerivativeSumMaterial
    property_name = Fa2
    sum_materials = 'Fa'
    coupled_variables = 'c'
    outputs = exodus
  [../]

  # derivatives declared after being used
  [./sum_b_1]
    type = DerivativeSumMaterial
    property_name = Fb1
    sum_materials = 'Fb'
    coupled_variables = 'c'
    outputs = exodus
  [../]
  [./free_energy_b]
    type = DerivativeParsedMaterial
    property_name = Fb
    coupled_variables = 'c'
    expression = 'c^4'
  [../]

  # derivatives declared before being used
  [./free_energy_c]
    type = DerivativeParsedMaterial
    property_name = Fc
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_c_2]
    type = DerivativeSumMaterial
    property_name = Fc2
    sum_materials = 'Fc'
    coupled_variables = 'c'
    outputs = exodus
  [../]

  # non-existing derivatives
  [./free_energy_d]
    type = ParsedMaterial
    property_name = Fd
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_d_1]
    type = DerivativeSumMaterial
    property_name = Fd1
    sum_materials = 'Fd'
    coupled_variables = 'c'
    outputs = exodus
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'NEWTON'
  num_steps = 1
  dt = 1e-5
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/phase_field/tutorials/spinodal_decomposition/s4_mobility.i)

#
# Example simulation of an iron-chromium alloy at 500 C. Equilibrium
# concentrations are at 23.6 and 82.3 mol% Cr. Kappa value, free energy equation,
# and mobility equation were provided by Lars Hoglund. Solved using the split
# form of the Cahn-Hilliard equation.
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  elem_type = QUAD4
  nx = 25
  ny = 25
  nz = 0
  xmin = 0
  xmax = 25
  ymin = 0
  ymax = 25
  zmin = 0
  zmax = 0
  uniform_refine = 2
[]

[Variables]
  [./c]   # Mole fraction of Cr (unitless)
    order = FIRST
    family = LAGRANGE
  [../]
  [./w]   # Chemical potential (eV/mol)
    order = FIRST
    family = LAGRANGE
  [../]
[]

[ICs]
  [./concentrationIC]   # 46.774 mol% Cr with variations
    type = RandomIC
    min = 0.44774
    max = 0.48774
    seed = 210
    variable = c
  [../]
[]

[BCs]
  [./Periodic]
    [./c_bcs]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Kernels]
  [./w_dot]
    variable = w
    v = c
    type = CoupledTimeDerivative
  [../]
  [./coupled_res]
    variable = w
    type = SplitCHWRes
    mob_name = M
  [../]
  [./coupled_parsed]
    variable = c
    type = SplitCHParsed
    f_name = f_loc
    kappa_name = kappa_c
    w = w
  [../]
[]

[Materials]
  # d is a scaling factor that makes it easier for the solution to converge
  # without changing the results. It is defined in each of the first three
  # materials and must have the same value in each one.
  [./kappa]                  # Gradient energy coefficient (eV nm^2/mol)
    type = GenericFunctionMaterial
    prop_names = 'kappa_c'
    prop_values = '8.125e-16*6.24150934e+18*1e+09^2*1e-27'
                  # kappa_c *eV_J*nm_m^2* d
  [../]
  [./mobility]               # Mobility (nm^2 mol/eV/s)
    # NOTE: This is a fitted equation, so only 'Conv' has units
    type = DerivativeParsedMaterial
    property_name = M
    coupled_variables = c
    constant_names =       'Acr    Bcr    Ccr    Dcr
                            Ecr    Fcr    Gcr
                            Afe    Bfe    Cfe    Dfe
                            Efe    Ffe    Gfe
                            nm_m   eV_J   d'
    constant_expressions = '-32.770969 -25.8186669 -3.29612744 17.669757
                            37.6197853 20.6941796  10.8095813
                            -31.687117 -26.0291774 0.2286581   24.3633544
                            44.3334237 8.72990497  20.956768
                            1e+09      6.24150934e+18          1e-27'
    expression = 'nm_m^2/eV_J/d*((1-c)^2*c*10^
                (Acr*c+Bcr*(1-c)+Ccr*c*log(c)+Dcr*(1-c)*log(1-c)+
                Ecr*c*(1-c)+Fcr*c*(1-c)*(2*c-1)+Gcr*c*(1-c)*(2*c-1)^2)
                +c^2*(1-c)*10^
                (Afe*c+Bfe*(1-c)+Cfe*c*log(c)+Dfe*(1-c)*log(1-c)+
                Efe*c*(1-c)+Ffe*c*(1-c)*(2*c-1)+Gfe*c*(1-c)*(2*c-1)^2))'
    derivative_order = 1
    outputs = exodus
  [../]
  [./local_energy]           # Local free energy function (eV/mol)
    type = DerivativeParsedMaterial
    property_name = f_loc
    coupled_variables = c
    constant_names = 'A   B   C   D   E   F   G  eV_J  d'
    constant_expressions = '-2.446831e+04 -2.827533e+04 4.167994e+03 7.052907e+03
                            1.208993e+04 2.568625e+03 -2.354293e+03
                            6.24150934e+18 1e-27'
    expression = 'eV_J*d*(A*c+B*(1-c)+C*c*log(c)+D*(1-c)*log(1-c)+
                E*c*(1-c)+F*c*(1-c)*(2*c-1)+G*c*(1-c)*(2*c-1)^2)'
    derivative_order = 2
  [../]
  [./precipitate_indicator]  # Returns 1/625 if precipitate
    type = ParsedMaterial
    property_name = prec_indic
    coupled_variables = c
    expression = if(c>0.6,0.0016,0)
  [../]
[]

[Postprocessors]
  [./step_size]             # Size of the time step
    type = TimestepSize
  [../]
  [./iterations]            # Number of iterations needed to converge timestep
    type = NumNonlinearIterations
  [../]
  [./nodes]                 # Number of nodes in mesh
    type = NumNodes
  [../]
  [./evaluations]           # Cumulative residual calculations for simulation
    type = NumResidualEvaluations
  [../]
  [./precipitate_area]      # Fraction of surface devoted to precipitates
    type = ElementIntegralMaterialProperty
    mat_prop = prec_indic
  [../]
  [./active_time]           # Time computer spent on simulation
    type = PerfGraphData
    section_name = "Root"
    data_type = total
  [../]
[]

[Preconditioning]
  [./coupled]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  l_max_its = 30
  l_tol = 1e-6
  nl_max_its = 50
  nl_abs_tol = 1e-9
  end_time = 604800   # 7 days
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type
                         -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly
                         ilu          1'
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    cutback_factor = 0.8
    growth_factor = 1.5
    optimal_iterations = 7
  [../]
  [./Adaptivity]
    coarsen_fraction = 0.1
    refine_fraction = 0.7
    max_h_level = 2
  [../]
[]

[Debug]
  show_var_residual_norms = true
[]

[Outputs]
  exodus = true
  console = true
  csv = true
  [./console]
    type = Console
    max_rows = 10
  [../]
[]

(modules/heat_transfer/test/tests/convective_heat_flux/coupled.i)

[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 10
[]

[Variables]
  [./temp]
    initial_condition = 200.0
  [../]
[]

[Kernels]
  [./heat_dt]
    type = TimeDerivative
    variable = temp
  [../]
  [./heat_conduction]
    type = Diffusion
    variable = temp
  [../]
  [./heat]
    type = BodyForce
    variable = temp
    value = 0
  [../]
[]

[BCs]
  [./right]
    type = ConvectiveHeatFluxBC
    variable = temp
    boundary = 'right'
    T_infinity = T_inf
    heat_transfer_coefficient = htc
    heat_transfer_coefficient_dT = dhtc_dT
  [../]
[]

[Materials]
  [./T_inf]
    type = ParsedMaterial
    property_name = T_inf
    coupled_variables = temp
    expression = 'temp + 1'
  [../]
  [./htc]
    type = ParsedMaterial
    property_name = htc
    coupled_variables = temp
    expression = 'temp / 100 + 1'
  [../]
  [./dhtc_dT]
    type = ParsedMaterial
    property_name = dhtc_dT
    coupled_variables = temp
    expression = '1 / 100'
  [../]
[]

[Postprocessors]
  [./left_temp]
    type = SideAverageValue
    variable = temp
    boundary = left
    execute_on = 'TIMESTEP_END initial'
  [../]
  [./right_temp]
    type = SideAverageValue
    variable = temp
    boundary = right
  [../]
  [./right_flux]
    type = SideDiffusiveFluxAverage
    variable = temp
    boundary = right
    diffusivity = 1
  [../]
[]

[Executioner]
  type = Transient

  num_steps = 10
  dt = 1

  nl_abs_tol = 1e-12
[]

[Outputs]
  [./out]
    type = CSV
    time_step_interval = 10
  [../]
[]

(modules/thermal_hydraulics/tutorials/single_phase_flow/06_custom_closures.i)

T_in = 300. # K
m_dot_in = 1e-2 # kg/s
press = 10e5 # Pa

# core parameters
core_length = 1. # m
core_n_elems = 25
core_dia = '${units 2. cm -> m}'
core_pitch = '${units 8.7 cm -> m}'
A_core = '${fparse core_pitch^2 - 0.25 *pi * core_dia^2}'
P_wet_core = '${fparse 4*core_pitch + pi * core_dia}'
Dh_core = '${fparse 4 * A_core / P_wet_core}'

# pipe parameters
pipe_dia = '${units 10. cm -> m}'
A_pipe = '${fparse 0.25 * pi * pipe_dia^2}'

tot_power = 2000 # W

# heat exchanger parameters
hx_dia_inner = '${units 12. cm -> m}'
hx_wall_thickness = '${units 5. mm -> m}'
hx_dia_outer = '${units 50. cm -> m}'
hx_radius_wall = '${fparse hx_dia_inner / 2. + hx_wall_thickness}'
hx_length = 1.5 # m
hx_n_elems = 25

m_dot_sec_in = 1. # kg/s

[GlobalParams]
  initial_p = ${press}
  initial_vel = 0.0001
  initial_T = ${T_in}
  initial_vel_x = 0
  initial_vel_y = 0
  initial_vel_z = 0
  gravity_vector = '0 0 0'

  rdg_slope_reconstruction = minmod
  scaling_factor_1phase = '1 1e-2 1e-4'
  scaling_factor_rhoV = 1
  scaling_factor_rhouV = 1e-2
  scaling_factor_rhovV = 1e-2
  scaling_factor_rhowV = 1e-2
  scaling_factor_rhoEV = 1e-4
  closures = thm_closures
  fp = he
[]

[Functions]
  [m_dot_sec_fn]
    type = PiecewiseLinear
    xy_data = '
      0    0
      10 ${m_dot_sec_in}'
  []
[]

[FluidProperties]
  [he]
    type = IdealGasFluidProperties
    molar_mass = 4e-3
    gamma = 1.67
    k = 0.2556
    mu = 3.22639e-5
  []

  [water]
    type = StiffenedGasFluidProperties
    gamma = 2.35
    cv = 1816.0
    q = -1.167e6
    p_inf = 1.0e9
    q_prime = 0
  []
[]

[Closures]
  [thm_closures]
    type = Closures1PhaseTHM
  []
  [none_closures]
    type = Closures1PhaseNone
  []
[]

[Materials]
  [Re_mat]
    type = ADReynoldsNumberMaterial
    Re = Re
    rho = rho
    vel = vel
    D_h = D_h
    mu = mu
    block = hx/pri
  []
  [f_mat]
    type = ADParsedMaterial
    property_name = f_D
    constant_names = 'a b c'
    constant_expressions = '1 0.1 -0.5'
    material_property_names = 'Re'
    expression = 'a + b * Re^c'
    block = hx/pri
  []
  [Pr_mat]
    type = ADPrandtlNumberMaterial
    Pr = Pr
    cp = cp
    mu = mu
    k = k
    block = hx/pri
  []
  [Nu_mat]
    type = ADParsedMaterial
    property_name = 'Nu'
    constant_names = 'a b c'
    constant_expressions = '0.03 0.9 0.5'
    material_property_names = 'Re Pr'
    expression = 'a * Re ^b * Pr^c'
    block = hx/pri
  []
  [Hw_mat]
    type = ADConvectiveHeatTransferCoefficientMaterial
    D_h = D_h
    k = k
    Nu = Nu
    Hw = Hw
    block = hx/pri
  []
[]
[SolidProperties]
  [steel]
    type = ThermalFunctionSolidProperties
    rho = 8050
    k = 45
    cp = 466
  []
[]

[Components]
  [total_power]
    type = TotalPower
    power = ${tot_power}
  []
  [up_pipe_1]
    type = FlowChannel1Phase
    position = '0 0 0'
    orientation = '0 0 1'
    length = 0.5
    n_elems = 15
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [jct1]
    type = JunctionParallelChannels1Phase
    position = '0 0 0.5'
    connections = 'up_pipe_1:out core_chan:in'
    volume = 1e-5
  []
  [core_chan]
    type = FlowChannel1Phase
    position = '0 0 0.5'
    orientation = '0 0 1'
    length = ${core_length}
    n_elems = ${core_n_elems}
    roughness = .0001
    A = ${A_core}
    D_h = ${Dh_core}
  []

  [core_hs]
    type = HeatStructureCylindrical
    position = '0 0 0.5'
    orientation = '0 0 1'
    length = ${core_length}
    n_elems = ${core_n_elems}
    names = 'block'
    widths = '${fparse core_dia / 2.}'
    solid_properties = 'steel'
    solid_properties_T_ref = '300'
    n_part_elems = 3
  []

  [core_heating]
    type = HeatSourceFromTotalPower
    hs = core_hs
    regions = block
    power = total_power
  []

  [core_ht]
    type = HeatTransferFromHeatStructure1Phase
    flow_channel = core_chan
    hs = core_hs
    hs_side = outer
    P_hf = '${fparse pi * core_dia}'
  []

  [jct2]
    type = JunctionParallelChannels1Phase
    position = '0 0 1.5'
    connections = 'core_chan:out up_pipe_2:in'
    volume = 1e-5
  []

  [up_pipe_2]
    type = FlowChannel1Phase
    position = '0 0 1.5'
    orientation = '0 0 1'
    length = 0.5
    n_elems = 10
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [jct3]
    type = JunctionOneToOne1Phase
    connections = 'up_pipe_2:out top_pipe_1:in'
  []

  [top_pipe_1]
    type = FlowChannel1Phase
    position = '0 0 2'
    orientation = '1 0 0'
    length = 0.5
    n_elems = 10
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [top_pipe_2]
    type = FlowChannel1Phase
    position = '0.5 0 2'
    orientation = '1 0 0'
    length = 0.5
    n_elems = 10
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [jct4]
    type = VolumeJunction1Phase
    position = '0.5 0 2'
    volume = 1e-5
    connections = 'top_pipe_1:out top_pipe_2:in press_pipe:in'
  []

  [press_pipe]
    type = FlowChannel1Phase
    position = '0.5 0 2'
    orientation = '0 1 0'
    length = 0.2
    n_elems = 5
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [pressurizer]
    type = InletStagnationPressureTemperature1Phase
    p0 = ${press}
    T0 = ${T_in}
    input = press_pipe:out
  []

  [jct5]
    type = JunctionOneToOne1Phase
    connections = 'top_pipe_2:out down_pipe_1:in'
  []

  [down_pipe_1]
    type = FlowChannel1Phase
    position = '1 0 2'
    orientation = '0 0 -1'
    length = 0.25
    A = ${A_pipe}
    n_elems = 5
  []

  [jct6]
    type = JunctionParallelChannels1Phase
    position = '1 0 1.75'
    connections = 'down_pipe_1:out hx/pri:in'
    volume = 1e-5
  []

  [hx]
    [pri]
      type = FlowChannel1Phase
      position = '1 0 1.75'
      orientation = '0 0 -1'
      length = ${hx_length}
      n_elems = ${hx_n_elems}
      roughness = 1e-5
      A = '${fparse pi * hx_dia_inner * hx_dia_inner / 4.}'
      D_h = ${hx_dia_inner}
      closures = none_closures
    []

    [ht_pri]
      type = HeatTransferFromHeatStructure1Phase
      hs = hx/wall
      hs_side = inner
      flow_channel = hx/pri
      P_hf = '${fparse pi * hx_dia_inner}'
    []

    [wall]
      type = HeatStructureCylindrical
      position = '1 0 1.75'
      orientation = '0 0 -1'
      length = ${hx_length}
      n_elems = ${hx_n_elems}
      widths = '${hx_wall_thickness}'
      n_part_elems = '3'
      solid_properties = 'steel'
      solid_properties_T_ref = '300'
      names = '0'
      inner_radius = '${fparse hx_dia_inner / 2.}'
    []

    [ht_sec]
      type = HeatTransferFromHeatStructure1Phase
      hs = hx/wall
      hs_side = outer
      flow_channel = hx/sec
      P_hf = '${fparse 2 * pi * hx_radius_wall}'
    []

    [sec]
      type = FlowChannel1Phase
      position = '${fparse 1 + hx_wall_thickness} 0 0.25'
      orientation = '0 0 1'
      length = ${hx_length}
      n_elems = ${hx_n_elems}
      A = '${fparse pi * (hx_dia_outer * hx_dia_outer / 4. - hx_radius_wall * hx_radius_wall)}'
      D_h = '${fparse hx_dia_outer - (2 * hx_radius_wall)}'
      fp = water
      initial_T = 300
    []
  []

  [jct7]
    type = JunctionParallelChannels1Phase
    position = '1 0 0.5'
    connections = 'hx/pri:out down_pipe_2:in'
    volume = 1e-5
  []

  [down_pipe_2]
    type = FlowChannel1Phase
    position = '1 0 0.25'
    orientation = '0 0 -1'
    length = 0.25
    n_elems = 10
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [jct8]
    type = JunctionOneToOne1Phase
    connections = 'down_pipe_2:out bottom_1:in'
  []

  [bottom_1]
    type = FlowChannel1Phase
    position = '1 0 0'
    orientation = '-1 0 0'
    length = 0.5
    n_elems = 5
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [pump]
    type = Pump1Phase
    position = '0.5 0 0'
    connections = 'bottom_1:out bottom_2:in'
    volume = 1e-4
    A_ref = ${A_pipe}
    head = 0
  []

  [bottom_2]
    type = FlowChannel1Phase
    position = '0.5 0 0'
    orientation = '-1 0 0'
    length = 0.5
    n_elems = 5
    A = ${A_pipe}
    D_h = ${pipe_dia}
  []

  [jct9]
    type = JunctionOneToOne1Phase
    connections = 'bottom_2:out up_pipe_1:in'
  []

  [inlet_sec]
    type = InletMassFlowRateTemperature1Phase
    input = 'hx/sec:in'
    m_dot = 0
    T = 300
  []

  [outlet_sec]
    type = Outlet1Phase
    input = 'hx/sec:out'
    p = 1e5
  []
[]

[ControlLogic]
  [set_point]
    type = GetFunctionValueControl
    function = ${m_dot_in}
  []

  [pid]
    type = PIDControl
    initial_value = 0.0
    set_point = set_point:value
    input = m_dot_pump
    K_p = 1.
    K_i = 4.
    K_d = 0
  []

  [set_pump_head]
    type = SetComponentRealValueControl
    component = pump
    parameter = head
    value = pid:output
  []

  [m_dot_sec_inlet_ctrl]
    type = GetFunctionValueControl
    function = m_dot_sec_fn
  []

  [set_m_dot_sec_ctrl]
    type = SetComponentRealValueControl
    component = inlet_sec
    parameter = m_dot
    value = m_dot_sec_inlet_ctrl:value
  []
[]

[Postprocessors]
  [power_to_coolant]
    type = ADHeatRateConvection1Phase
    block = core_chan
    P_hf = '${fparse pi *core_dia}'
  []

  [m_dot_pump]
    type = ADFlowJunctionFlux1Phase
    boundary = core_chan:in
    connection_index = 1
    equation = mass
    junction = jct7
  []

  [core_T_out]
    type = SideAverageValue
    boundary = core_chan:out
    variable = T
  []

  [core_p_in]
    type = SideAverageValue
    boundary = core_chan:in
    variable = p
  []

  [core_p_out]
    type = SideAverageValue
    boundary = core_chan:out
    variable = p
  []

  [core_delta_p]
    type = ParsedPostprocessor
    pp_names = 'core_p_in core_p_out'
    expression = 'core_p_in - core_p_out'
  []

  [hx_pri_T_out]
    type = SideAverageValue
    boundary = hx/pri:out
    variable = T
  []

  [hx_sec_T_in]
    type = SideAverageValue
    boundary = inlet_sec
    variable = T
  []

  [hx_sec_T_out]
    type = SideAverageValue
    boundary = outlet_sec
    variable = T
  []
  [m_dot_sec]
    type = ADFlowBoundaryFlux1Phase
    boundary = inlet_sec
    equation = mass
  []

  [Hw_hx_pri]
    type = ADElementAverageMaterialProperty
    mat_prop = Hw
    block = hx/pri
  []
  [fD_hx_pri]
    type = ADElementAverageMaterialProperty
    mat_prop = f_D
    block = hx/pri
  []

[]

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

[Executioner]
  type = Transient
  start_time = 0

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1
  []
  dtmax = 5
  end_time = 500

  line_search = basic
  solve_type = NEWTON

  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'

  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8
  nl_max_its = 25

[]

[Outputs]
  exodus = true

  [console]
    type = Console
    max_rows = 1
    outlier_variable_norms = false
  []
  print_linear_residuals = false
[]

(tutorials/tutorial02_multiapps/step03_coupling/01_parent.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
  []
[]

[AuxVariables]
  [vt]
  []
[]

[Kernels]
  [diff]
    type = MatDiffusion
    variable = u
  []
  [force]
    type = BodyForce
    variable = u
    value = 1.
  []
  [td]
    type = TimeDerivative
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Materials]
  [diff]
    type = ParsedMaterial
    property_name = D
    coupled_variables = 'vt'
    expression = 'vt'
  []
[]

[Executioner]
  type = Transient
  end_time = 2
  dt = 0.2

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

[MultiApps]
  [micro]
    type = TransientMultiApp
    positions = '0.15 0.15 0  0.45 0.45 0  0.75 0.75 0'
    input_files = '01_sub.i'
    execute_on = timestep_end
    output_in_position = true
  []
[]

[Transfers]
  [push_u]
    type = MultiAppVariableValueSampleTransfer
    to_multi_app = micro
    source_variable = u
    variable = ut
  []

  [pull_v]
    type = MultiAppPostprocessorInterpolationTransfer
    from_multi_app = micro
    variable = vt
    postprocessor = average_v
  []
[]

(modules/electromagnetics/test/tests/bcs/reflectionBC_helmholtz/ReflectionTest.i)

# problem: -(ku')' - c^2 * u' = 0 , 0 < x < L, u: R -> C
# u(x=0) = g0 , u'(x = L) = 2jcf(L)*exp(jcLf(L)) - jcf(L)u(x = L)
# c = a + jb , k = d + jh

[GlobalParams]
  coeff_real = 15
  coeff_imag = 7
[]

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = 1
    nx = 100
  []
[]

[Variables]
  [u_real]
    order = FIRST
    family = LAGRANGE
  []
  [u_imag]
    order = FIRST
    family = LAGRANGE
  []
[]

[Functions]
  [cos]
    type = ParsedFunction
    expression = 'cos(0.5)'
  []
[]

[Materials]
  [ASquaredMinusBSquared]
    type = ADParsedMaterial
    property_name = ASquaredMinusBSquared
    expression = '(15*15 - 7*7)'
  []
  [2TimesAB]
    type = ADParsedMaterial
    property_name = 2TimesAB
    expression = '2*15*7'
  []
  [negative_2TimesAB]
    type = ADParsedMaterial
    property_name = negative_2TimesAB
    expression = '-2*15*7'
  []
[]

[Kernels]
  [laplacian_real]
    type = Diffusion
    variable = u_real
  []
  [coeffField_real]
    type = ADMatReaction
    reaction_rate = ASquaredMinusBSquared
    variable = u_real
  []
  [coupledField_real]
    type = ADMatCoupledForce
    v = u_imag
    mat_prop_coef = negative_2TimesAB
    variable = u_real
  []
  [laplacian_imag]
    type = Diffusion
    variable = u_imag
  []
  [coeffField_imag]
    type = ADMatReaction
    reaction_rate = ASquaredMinusBSquared
    variable = u_imag
  []
  [coupledField_imag]
    type = ADMatCoupledForce
    v = u_real
    mat_prop_coef = 2TimesAB
    variable = u_imag
  []
[]

[BCs]
  [left_real]
    type = DirichletBC
    value = 0
    boundary = left
    variable = u_real
  []
  [left_imag]
    type = DirichletBC
    value = 1
    boundary = left
    variable = u_imag
  []
  [right_real]
    type = EMRobinBC
    func_real = cos
    boundary = right
    component = real
    variable = u_real
    field_real = u_real
    field_imaginary = u_imag
    sign = negative
  []
  [right_imag]
    type = EMRobinBC
    func_real = cos
    boundary = right
    component = imaginary
    variable = u_imag
    field_real = u_real
    field_imaginary = u_imag
    sign = negative
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/materials/derivative_material_interface/old_older.i)

#
# This test validates the correct application of the chain rule to coupled
# material properties within DerivativeParsedMaterials
#

[Mesh]
  type = GeneratedMesh
  dim = 1
[]

[Problem]
  solve = false
  kernel_coverage_check = false
[]

[Variables]
  [dummy]
  []
[]

[Materials]
  [t_square]
    type = GenericFunctionMaterial
    prop_names = t2
    prop_values = 't^2'
  []

  [t2_old]
    type = ParsedMaterial
    property_name = t2_old
    expression = t2_old
    material_property_names = 't2_old:=Old[t2]'
  []
  [t2_older]
    type = ParsedMaterial
    property_name = t2_older
    expression = t2_older
    material_property_names = 't2_older:=Older[t2]'
  []
[]

[Executioner]
  type = Transient
  dt = 1.5
  num_steps = 5
[]

[Postprocessors]
  [t2]
    type = ElementAverageMaterialProperty
    mat_prop = t2
  []
  [t2_old]
    type = ElementAverageMaterialProperty
    mat_prop = t2_old
  []
  [t2_older]
    type = ElementAverageMaterialProperty
    mat_prop = t2_older
  []
[]

[Outputs]
  csv = true
  print_linear_residuals = false
[]

(test/tests/materials/compile_time_derivative/test.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 2
    ny = 2
    nz = 2
  []
[]


[Problem]
  solve = false
[]

[Variables]
  [a]
    [InitialCondition]
      type = FunctionIC
      function = x
    []
  []
  [b]
    [InitialCondition]
      type = FunctionIC
      function = y
    []
  []
  [c]
    [InitialCondition]
      type = FunctionIC
      function = z
    []
  []
[]

[Materials]
  [ctd]
    type = CTDCoupledVarTest
    x = a
    y = b
    z = c
    property_name = F
  []
  [parsed]
    type = DerivativeParsedMaterial
    coupled_variables = 'a b c'
    expression = 'a^3*b^4*c^5 + sin(a)*cos(b)/(c+0.1) + log(a+0.1)*sin(b)*cos(c)'
    property_name = G
  []

  [L2_difference]
    type = ParsedMaterial
    expression = '(f0-g0)^2+(f1-g1)^2+(f2-g2)^2+(f3-g3)^2+(f4-g4)^2+(f5-g5)^2+(f6-g6)^2+(f7-g7)^2+(f8-g8)^2+(f9-g9)^2+(f10-g10)^2+(f11-g11)^2+(f12-g12)^2+(f13-g13)^2+(f14-g14)^2+(f15-g15)^2+(f16-g16)^2+(f17-g17)^2+(f18-g18)^2+(f19-g19)^2'
    material_property_names = 'f0:=F g0:=G f1:=dF/da g1:=dG/da f2:=dF/db g2:=dG/db f3:=dF/dc g3:=dG/dc f4:=d^2F/da^2 g4:=d^2G/da^2 f5:=d^2F/dadb g5:=d^2G/dadb f6:=d^2F/dadc g6:=d^2G/dadc f7:=d^2F/db^2 g7:=d^2G/db^2 f8:=d^2F/dbdc g8:=d^2G/dbdc f9:=d^2F/dc^2 g9:=d^2G/dc^2 f10:=d^3F/da^2db g10:=d^3G/da^2db f11:=d^3F/da^2dc g11:=d^3G/da^2dc f12:=d^3F/da^3 g12:=d^3G/da^3 f13:=d^3F/dadb^2 g13:=d^3G/dadb^2 f14:=d^3F/dadbdc g14:=d^3G/dadbdc f15:=d^3F/dadc^2 g15:=d^3G/dadc^2 f16:=d^3F/db^2dc g16:=d^3G/db^2dc f17:=d^3F/db^3 g17:=d^3G/db^3 f18:=d^3F/dbdc^2 g18:=d^3G/dbdc^2 f19:=d^3F/dc^3 g19:=d^3G/dc^3'
    property_name = L2
  []
[]

[Postprocessors]
  [L2_int]
    type = ElementIntegralMaterialProperty
    mat_prop = L2
  []
[]

[Executioner]
  type = Steady
[]

[Outputs]
  csv = true
[]

(modules/phase_field/test/tests/slkks/weighted_average.i)

#
# This tests the WeightedVariableAverage postprocessor, which averages a variable field
# with weights applied from a material property. This can be used to obtain average
# concentrations in different phases (based on the total physical concentration variable).
#
#

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
  []
[]

[Variables]
  [c]
    [./InitialCondition]
      type = FunctionIC
      function = x*0.5
    []
   []
   [eta]
     [./InitialCondition]
       type = FunctionIC
       function = x
     []
   []
 []

 [Materials]
   [h]
     type = ParsedMaterial
     coupled_variables = eta
     property_name = h
     expression = 'if(eta>0.5,1,0)'
   []
 []

 [Postprocessors]
   [c1]
     type = WeightedVariableAverage
     v = c
     weight = h
     execute_on = INITIAL
   []
 []

 [Problem]
   solve = false
 []

 [Executioner]
   type = Steady
 []

 [Outputs]
   csv = true
 []

(modules/combined/test/tests/phase_field_fracture/crack2d_aniso.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 20
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./All]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'strain_yy stress_yy'
        planar_formulation = PLANE_STRAIN
      [../]
    [../]
  [../]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
  [./off_disp]
    type = AllenCahnElasticEnergyOffDiag
    variable = c
    displacements = 'disp_x disp_y'
    mob_name = L
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.05 1e-6'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '127.0 70.8 70.8 127.0 70.8 127.0 73.55 73.55 73.55'
    fill_method = symmetric9
    euler_angle_1 = 30
    euler_angle_2 = 0
    euler_angle_3 = 0
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = stress_spectral
    use_current_history_variable = true
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '1.0e-6'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./av_stress_yy]
    type = ElementAverageValue
    variable = stress_yy
  [../]
  [./av_strain_yy]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solving_package'
  petsc_options_value = 'lu superlu_dist'

  nl_rel_tol = 1e-8
  l_tol = 1e-4
  l_max_its = 100
  nl_max_its = 10

  dt = 5e-5
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/fsi/test/tests/2d-finite-strain-steady/thermal-me.i)

# Units: specific_heat_capacity--cp--J/(kg.K); density--rho--kg/(cm^3);
# dynamic_viscosity--mu--kg/(cm.s); thermal_conductivity--k--W/(cm.K);
# pressure--kg/(cm.s^2); force--kg.cm/s^2

outlet_pressure = 0
inlet_velocity = 150 # cm/s
ini_temp = 593 # K
heat_transfer_coefficient = 9 # W/(cm2.K)
g = -981 # cm/s2
alpha_fluid = 2e-4 # thermal expansion coefficient of fluid used in INSADBoussinesqBodyForce

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  file = '2layers_2d_midline.msh'
[]

[Variables]
  [velocity]
    family = LAGRANGE_VEC
    order = FIRST
    block = 'fluid'
  []
  [p]
    family = LAGRANGE
    order = FIRST
    block = 'fluid'
  []
  [Tf]
    family = LAGRANGE
    order = FIRST
    block = 'fluid'
  []
  [Ts]
    family = LAGRANGE
    order = FIRST
    block = 'solid'
  []
  [disp_x]
    family = LAGRANGE
    order = FIRST
    block = 'solid fluid'
  []
  [disp_y]
    family = LAGRANGE
    order = FIRST
    block = 'solid fluid'
  []
[]

[AuxVariables]
  [heat_source]
    family = MONOMIAL
    order = FIRST
    block = 'solid'
  []
[]

[ICs]
  [initial_velocity]
    type = VectorConstantIC
    variable = velocity
    x_value = 0
    y_value = ${inlet_velocity}
    z_value = 0
  []
  [initial_p]
    type = FunctionIC
    variable = p
    function = ini_p
  []
  [initial_Tf]
    type = ConstantIC
    variable = Tf
    value = ${ini_temp}
  []
  [initial_Ts]
    type = ConstantIC
    variable = Ts
    value = ${ini_temp}
  []
[]

[Kernels]
  [fluid_mass]
    type = INSADMass
    variable = p
    use_displaced_mesh = true
  []
  [fluid_mass_pspg]
    type = INSADMassPSPG
    variable = p
    use_displaced_mesh = true
  []
  [fluid_momentum_time]
    type = INSADMomentumTimeDerivative
    variable = velocity
    use_displaced_mesh = true
  []
  [fluid_momentum_convection]
    type = INSADMomentumAdvection
    variable = velocity
    use_displaced_mesh = true
  []
  [fluid_momentum_viscous]
    type = INSADMomentumViscous
    variable = velocity
    use_displaced_mesh = true
  []
  [fluid_momentum_pressure]
    type = INSADMomentumPressure
    variable = velocity
    pressure = p
    integrate_p_by_parts = true
    use_displaced_mesh = true
  []
  [fluid_momentum_gravity]
    type = INSADGravityForce
    variable = velocity
    gravity = '0 ${g} 0'
    use_displaced_mesh = true
  []
  [fluid_momentum_buoyancy]
    type = INSADBoussinesqBodyForce
    variable = velocity
    gravity = '0 ${g} 0'
    alpha_name = 'alpha_fluid'
    ref_temp = 'T_ref'
    temperature = Tf
    use_displaced_mesh = true
  []
  [fluid_momentum_supg]
    type = INSADMomentumSUPG
    variable = velocity
    velocity = velocity
    use_displaced_mesh = true
  []
  [fluid_temperature_time]
    type = INSADHeatConductionTimeDerivative
    variable = Tf
    use_displaced_mesh = true
  []
  [fluid_temperature_conduction]
    type = ADHeatConduction
    variable = Tf
    thermal_conductivity = 'k'
    use_displaced_mesh = true
  []
  [fluid_temperature_advection]
    type = INSADEnergyAdvection
    variable = Tf
    use_displaced_mesh = true
  []
  [fluid_temperature_supg]
    type = INSADEnergySUPG
    variable = Tf
    velocity = velocity
    use_displaced_mesh = true
  []

  [solid_temperature_time]
    type = ADHeatConductionTimeDerivative
    variable = Ts
    density_name = 'rho'
    specific_heat = 'cp'
    block = 'solid'
    use_displaced_mesh = true
  []
  [solid_temperature_conduction]
    type = ADHeatConduction
    variable = Ts
    thermal_conductivity = 'k'
    block = 'solid'
    use_displaced_mesh = true
  []
  [heat_source]
    type = ADCoupledForce
    variable = Ts
    v = heat_source
    block = 'solid'
    use_displaced_mesh = true
  []

  [disp_x_smooth]
    type = Diffusion
    variable = disp_x
    block = fluid
  []
  [disp_y_smooth]
    type = Diffusion
    variable = disp_y
    block = fluid
  []
[]

[Modules/TensorMechanics/Master]
  strain = FINITE
  material_output_order = FIRST
  generate_output = 'vonmises_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
  [solid]
    block = 'solid'
    temperature = Ts
    automatic_eigenstrain_names = true
  []
[]

[InterfaceKernels]
  [convection_heat_transfer]
    type = ConjugateHeatTransfer
    variable = Tf
    T_fluid = Tf
    neighbor_var = 'Ts'
    boundary = 'solid_wall'
    htc = 'htc'
    use_displaced_mesh = true
  []
[]

[AuxKernels]
  [heat_source_distribution_auxk]
    type = FunctionAux
    variable = heat_source
    function = heat_source_distribution_function
    block = 'solid'
    use_displaced_mesh = true
    execute_on = 'INITIAL TIMESTEP_BEGIN'
  []
[]

[BCs]
  [no_slip]
    type = VectorFunctionDirichletBC
    variable = velocity
    boundary = 'solid_wall'
    use_displaced_mesh = true
  []
  [inlet_velocity]
    type = VectorFunctionDirichletBC
    variable = velocity
    boundary = 'fluid_bottom'
    function_y = ${inlet_velocity}
    use_displaced_mesh = true
  []
  [symmetry]
    type = ADVectorFunctionDirichletBC
    variable = velocity
    boundary = 'fluid_wall'
    function_x = 0
    set_x_comp = true
    set_y_comp = false
    set_z_comp = false
    use_displaced_mesh = true
  []
  [outlet_p]
    type = DirichletBC
    variable = p
    boundary = 'fluid_top'
    value = ${outlet_pressure}
    use_displaced_mesh = true
  []
  [inlet_T]
    type = DirichletBC
    variable = Tf
    boundary = 'fluid_bottom'
    value = ${ini_temp}
    use_displaced_mesh = true
  []

  [pin1_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'pin1'
    value = 0
    use_displaced_mesh = true
  []
  [pin1_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'pin1'
    value = 0
    use_displaced_mesh = true
  []
  [top_and_bottom_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'solid_bottom solid_top fluid_top fluid_bottom'
    value = 0
    use_displaced_mesh = true
  []
  [left_and_right_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'fluid_wall fluid_bottom'
    value = 0
    use_displaced_mesh = true
  []
[]

[Materials]
  [rho_solid]
    type = ADParsedMaterial
    property_name = rho
    expression = '0.0110876 * pow(9.9672e-1 + 1.179e-5 * Ts - 2.429e-9 * pow(Ts,2) + 1.219e-12 * pow(Ts,3),-3)'
    coupled_variables = 'Ts'
    block = 'solid'
    use_displaced_mesh = true
  []
  [cp_solid]
    type = ADParsedMaterial
    property_name = cp
    expression = '0.76 * ((302.27 * pow((548.68 / Ts),2) * exp(548.68 / Ts)) / pow((exp(548.68 / Ts) - 1),2) + 2 * 8.463e-3 * Ts + 8.741e7 * 18531.7 * exp(-18531.7 / Ts) / pow(Ts,2)) + 0.24 * ((322.49 * pow((587.41/Ts),2) * exp(587.41 / Ts)) / pow((exp(587.41 / Ts) - 1),2) + 2 * 1.4679e-2 * Ts)'
    coupled_variables = 'Ts'
    block = 'solid'
    use_displaced_mesh = true
  []
  [k_solid]
    type = ADParsedMaterial
    property_name = k
    expression = '1.158/(7.5408 + 17.692 * (Ts / 1000) + 3.6142 * pow((Ts/1000),2)) + 74.105 * pow((Ts / 1000),-2.5) * exp(-16.35 / (Ts / 1000))'
    coupled_variables = 'Ts'
    block = 'solid'
    use_displaced_mesh = true
  []

  [rho_fluid]
    type = ADParsedMaterial
    property_name = rho
    expression = '(11096 - 1.3236 * Tf) * 1e-6'
    coupled_variables = 'Tf'
    block = 'fluid'
    use_displaced_mesh = true
  []
  [cp_fluid]
    type = ADParsedMaterial
    property_name = cp
    expression = '159 - 2.72e-2 * Tf + 7.12e-6 * pow(Tf,2)'
    coupled_variables = 'Tf'
    block = 'fluid'
    use_displaced_mesh = true
  []
  [k_fluid]
    type = ADParsedMaterial
    property_name = k
    expression = '(3.61 + 1.517e-2 * Tf - 1.741e-6 * pow(Tf,2)) * 1e-2'
    coupled_variables = 'Tf'
    block = 'fluid'
    use_displaced_mesh = true
  []
  [mu_fluid]
    type = ADParsedMaterial
    property_name = mu
    expression = '4.94e-6 * exp(754.1/Tf)'
    coupled_variables = 'Tf'
    block = 'fluid'
    use_displaced_mesh = true
  []
  [buoyancy_thermal_expansion_coefficient_fluid]
    type = ADGenericConstantMaterial
    prop_names = 'alpha_fluid'
    prop_values = '${alpha_fluid}'
    block = 'fluid'
    use_displaced_mesh = true
  []
  [buoyancy_reference_temperature_fluid]
    type = GenericConstantMaterial
    prop_names = 'T_ref'
    prop_values = '${ini_temp}'
    block = 'fluid'
    use_displaced_mesh = true
  []

  [ins_mat_fluid]
    type = INSADStabilized3Eqn
    velocity = velocity
    pressure = p
    temperature = Tf
    block = 'fluid'
    use_displaced_mesh = true
  []

  [htc]
    type = ADGenericFunctionMaterial
    prop_names = htc
    prop_values = htc_function
    use_displaced_mesh = true
  []

  [elasticity_solid]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 2e7
    poissons_ratio = 0.32
    block = 'solid'
    use_displaced_mesh = true
  []
  [thermal_expansion_solid]
    type = ComputeThermalExpansionEigenstrain
    temperature = Ts
    thermal_expansion_coeff = 2e-4
    stress_free_temperature = 593
    eigenstrain_name = thermal_expansion
    block = 'solid'
    use_displaced_mesh = true
  []
  [stress_solid]
    type = ComputeFiniteStrainElasticStress
    block = 'solid'
  []
[]

[Functions]
  [htc_function]
    type = ParsedFunction
    expression = ${heat_transfer_coefficient}
  []
  [ini_p]
    type = ParsedFunction
    expression = '0.010302 * 981 * (10 - y)'
  []
  [heat_source_distribution_function]
    type = ParsedFunction
    expression = '300 * sin(pi * y / 10)'
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
    solve_type = 'PJFNK'
  []
[]

[Executioner]
  type = Transient
  end_time = 1e4

  solve_type = 'NEWTON'
  petsc_options = '-snes_converged_reason -ksp_converged_reason -snes_linesearch_monitor'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu       NONZERO'
  line_search = 'none'

  nl_max_its = 30
  l_max_its = 100
  automatic_scaling = true
  compute_scaling_once = true
  off_diagonals_in_auto_scaling = true
  dtmin = 1
  nl_abs_tol = 1e-12

  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 6
    growth_factor = 1.5
    dt = 1
  []
[]

[Outputs]
  [csv]
    type = CSV
    file_base = 'thermal-me'
    execute_on = 'final'
  []
[]

[Postprocessors]
  [average_solid_Ts]
    type = ElementAverageValue
    variable = Ts
    block = 'solid'
    use_displaced_mesh = true
  []
  [average_fluid_Tf]
    type = ElementAverageValue
    variable = Tf
    block = 'fluid'
    use_displaced_mesh = true
  []
  [max_solid_Ts]
    type = ElementExtremeValue
    variable = Ts
    value_type = max
    block = 'solid'
    use_displaced_mesh = true
  []
  [max_fluid_Tf]
    type = ElementExtremeValue
    variable = Tf
    value_type = max
    block = 'fluid'
    use_displaced_mesh = true
  []
  [min_solid_Ts]
    type = ElementExtremeValue
    variable = Ts
    value_type = min
    block = 'solid'
    use_displaced_mesh = true
  []
  [min_fluid_Tf]
    type = ElementExtremeValue
    variable = Tf
    value_type = min
    block = 'fluid'
    use_displaced_mesh = true
  []
[]

[Debug]
  show_var_residual_norms = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_iso.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = strain_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/electromagnetics/test/tests/kernels/scalar_complex_helmholtz/scalar_complex_helmholtz.i)

# problem: -(cu')' - k^2 * u = -F , 0 < x < L, u: R -> C
# u(x=0) = g0 , u(x=L) = gL
# k = a + jb
#     a = a(x) =  2 * (1 + x/L)
#     b = b(x) =      (1 + x/L)
# c = d + jh
#     d = d(x) = 12 * (1 + x/L)^2
#     h = h(x) =  4 * (1 + x/L)^2
# L = 10

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = 10
    nx = 100
  []
[]

[Variables]
  [u_real]
    order = FIRST
    family = LAGRANGE
  []
  [u_imag]
    order = FIRST
    family = LAGRANGE
  []
[]

[Functions]
  [k_real]
    type = ParsedFunction
    expression = '2*(1 + x/10)'
  []
  [k_imag]
    type = ParsedFunction
    expression = '(1 + x/10)'
  []
  [d_func]
    type = ParsedFunction
    expression = '12 * (1 + x/10)^2'
  []
  [h_func]
    type = ParsedFunction
    expression = '4 * (1 + x/10)^2'
  []
  [negative_h_func]
    type = ParsedFunction
    expression = '-4 * (1 + x/10)^2'
  []
  [RHS_real]
    type = MMSTestFunc
    L = 10
    g0_real = 1
    g0_imag = -1
    gL_real = 0
    gL_imag = 0
    component = real
  []
  [RHS_imag]
    type = MMSTestFunc
    L = 10
    g0_real = 1
    g0_imag = -1
    gL_real = 0
    gL_imag = 0
    component = imaginary
  []
[]

[Materials]
  [k_real_mat]
    type = ADGenericFunctionMaterial
    prop_names = k_real_mat
    prop_values = k_real
  []
  [k_imag_mat]
    type = ADGenericFunctionMaterial
    prop_names = k_imag_mat
    prop_values = k_imag
  []
  [wave_equation_coefficient]
    type = WaveEquationCoefficient
    k_real = k_real_mat
    k_imag = k_imag_mat
    eps_rel_real = 1
    eps_rel_imag = 0
    mu_rel_real = 1
    mu_rel_imag = 0
  []
  [negative_wave_equation_coefficient_imaginary]
    type = ADParsedMaterial
    property_name = negative_wave_equation_coefficient_imaginary
    material_property_names = wave_equation_coefficient_imaginary
    expression = '-1 * wave_equation_coefficient_imaginary'
  []
[]

[Kernels]
  [laplacian_real]
    type = FunctionDiffusion
    function = d_func
    variable = u_real
  []
  [coupledLaplacian_real]
    type = FunctionDiffusion
    function = negative_h_func
    v = u_imag
    variable = u_real
  []
  [coeffField_real]
    type = ADMatReaction
    reaction_rate = wave_equation_coefficient_real
    variable = u_real
  []
  [coupledField_real]
    type = ADMatCoupledForce
    v = u_imag
    mat_prop_coef = negative_wave_equation_coefficient_imaginary
    variable = u_real
  []
  [bodyForce_real]
    type = BodyForce
    function = RHS_real
    variable = u_real
  []
  [laplacian_imag]
    type = FunctionDiffusion
    function = d_func
    variable = u_imag
  []
  [coupledLaplacian_imag]
    type = FunctionDiffusion
    function = h_func
    v = u_real
    variable = u_imag
  []
  [coeffField_imag]
    type = ADMatReaction
    reaction_rate = wave_equation_coefficient_real
    variable = u_imag
  []
  [coupledField_imag]
    type = ADMatCoupledForce
    v = u_real
    mat_prop_coef = wave_equation_coefficient_imaginary
    variable = u_imag
  []
  [bodyForce_imag]
    type = BodyForce
    function = RHS_imag
    variable = u_imag
  []
[]

[BCs]
  [left_real]
    type = DirichletBC
    value = 1
    boundary = left
    variable = u_real
  []
  [left_imag]
    type = DirichletBC
    value = -1
    boundary = left
    variable = u_imag
  []
  [right_real]
    type = DirichletBC
    value = 0
    boundary = right
    variable = u_real
  []
  [right_imag]
    type = DirichletBC
    value = 0
    boundary = right
    variable = u_imag
  []
[]

[Executioner]
  type = Steady
  solve_type = 'PJFNK'
[]

[Outputs]
  exodus = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_vi_solver.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 20
    xmax = 1
    ymax = 1
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[ICs]
  [./c_ic]
    type = FunctionIC
    function = ic
    variable = c
  [../]
[]

[Functions]
  [./ic]
    type = ParsedFunction
    expression = 'if(x<0.5 & y < 0.55 & y > 0.45,1, 0)'
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
  [./bounds_dummy]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = 'top bottom'
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = strain_spectral
    use_snes_vi_solver = true
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Bounds]
  [./c_upper_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = c
    bound_type = upper
    bound_value = 1.0
  [../]
  [./c_lower_bound]
    type = VariableOldValueBounds
    variable = bounds_dummy
    bounded_variable = c
    bound_type = lower
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type  -snes_type'
  petsc_options_value = 'lu vinewtonrsls'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/combined/test/tests/phase_field_fracture_viscoplastic/crack2d.i)

[Mesh]
  type = FileMesh
  file = crack_mesh.e
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
  volumetric_locking_correction = true
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./All]
        add_variables = true
        strain = Finite
        additional_generate_output = stress_yy
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = E_el
        mobility = L
        kappa = kappa_op
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
  [./stress_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./peeq]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
    use_displaced_mesh = true
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
    use_displaced_mesh = true
  [../]
[]

[AuxKernels]
  [./stress_yy]
    type = RankTwoAux
    variable = stress_yy
    rank_two_tensor = stress
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  [../]
  [./peeq]
    type = MaterialRealAux
    variable = peeq
    property = ep_eqv
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = 2
    function = '0.0001*t'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = 1
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = '1 2'
    value = 0
  [../]
[]

[UserObjects]
  [./flowstress]
    type = HEVPLinearHardening
    yield_stress = 300
    slope = 1000
    intvar_prop_name = ep_eqv
  [../]
  [./flowrate]
    type = HEVPFlowRatePowerLawJ2
    reference_flow_rate = 0.0001
    flow_rate_exponent = 10.0
    flow_rate_tol = 1
    strength_prop_name = flowstress
  [../]
  [./ep_eqv]
     type = HEVPEqvPlasticStrain
     intvar_rate_prop_name = ep_eqv_rate
  [../]
  [./ep_eqv_rate]
     type = HEVPEqvPlasticStrainRate
     flow_rate_prop_name = flowrate
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'l visco'
    prop_values = '0.08 1'
  [../]
  [./pfgc]
    type = GenericFunctionMaterial
    prop_names = 'gc_prop'
    prop_values = '1.0e-3'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./viscop_damage]
    type = HyperElasticPhaseFieldIsoDamage
    resid_abs_tol = 1e-18
    resid_rel_tol = 1e-8
    maxiters = 50
    max_substep_iteration = 5
    flow_rate_user_objects = 'flowrate'
    strength_user_objects = 'flowstress'
    internal_var_user_objects = 'ep_eqv'
    internal_var_rate_user_objects = 'ep_eqv_rate'
    numerical_stiffness = false
    damage_stiffness = 1e-8
    c = c
    F_name = E_el
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_grmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8

  l_max_its = 10
  nl_max_its = 10

  dt = 1
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/grain_growth/grain_growth_linearized_interface.i)

[GlobalParams]
  bound_value = 5.0
  op_num = 8
  var_name_base = phi
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 1000
  ymax = 1000
  nx = 100
  ny = 100
  uniform_refine = 1
[]

[Modules]
  [PhaseField]
    [GrainGrowthLinearizedInterface]
      op_name_base = gr
      mobility = L
      kappa = kappa_op
    []
  []
[]

[ICs]
  [PolycrystalICs]
    [PolycrystalColoringIC]
      polycrystal_ic_uo = RandomVoronoi
      nonlinear_preconditioning = true
    []
  []
[]

[UserObjects]
  [RandomVoronoi]
    type = PolycrystalVoronoi
    grain_num = 60
    int_width = 10
    rand_seed = 103838
  []
  [grain_tracker]
    type = GrainTracker
    threshold = -4.0
    compute_halo_maps = true # Only necessary for displaying HALOS
  []
[]

[AuxVariables]
  [unique_grains]
    order = CONSTANT
    family = MONOMIAL
  []
  [var_indices]
    order = CONSTANT
    family = MONOMIAL
  []
  [halos]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [unique_grains]
    type = FeatureFloodCountAux
    variable = unique_grains
    flood_counter = grain_tracker
    field_display = UNIQUE_REGION
    execute_on = 'initial timestep_end'
  []
  [var_indices]
    type = FeatureFloodCountAux
    variable = var_indices
    flood_counter = grain_tracker
    field_display = VARIABLE_COLORING
    execute_on = 'initial timestep_end'
  []
  [halos]
    type = FeatureFloodCountAux
    variable = halos
    flood_counter = grain_tracker
    field_display = HALOS
    execute_on = 'initial timestep_end'
  []
[]

[Materials]
  [properties]
    type = GenericConstantMaterial
    prop_names = 'gbmob gbenergy gbwidth gamma_asymm'
    prop_values = '100 6 10 1.5'
  []
  [kappa_op]
    type = ParsedMaterial
    material_property_names = 'gbenergy gbwidth'
    property_name = kappa_op
    expression = '3/4*gbenergy*gbwidth'
  []
  [L]
    type = ParsedMaterial
    material_property_names = 'gbmob gbwidth'
    property_name = L
    expression = '4/3*gbmob/gbwidth'
  []
  [mu]
    type = ParsedMaterial
    material_property_names = 'gbenergy gbwidth'
    property_name = mu
    expression = '6*gbenergy/gbwidth'
  []
[]

[Postprocessors]
  [dt]
    type = TimestepSize
    execute_on = 'initial TIMESTEP_END'
  []
[]

[BCs]
  [Periodic]
    [All]
      auto_direction = 'x y'
    []
  []
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK

  petsc_options_iname = '-pc_type -pc_hypre_type -snes_type'
  petsc_options_value = 'hypre    boomeramg      vinewtonrsls'

  l_tol = 1e-4
  nl_max_its = 10
  l_max_its = 45

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.02
    optimal_iterations = 6
  []
  end_time = 30
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/SinteringBase.i)

#input file to test the materials GrandPotentialTensorMaterial and GrandPotentialSinteringMaterial

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 17
  ny = 17
  xmin = 0
  xmax = 680
  ymin = 0
  ymax = 680
[]

[GlobalParams]
  op_num = 4
  var_name_base = gr
  int_width = 40
[]

[Variables]
  [./w]
  [../]
  [./phi]
  [../]
  [./PolycrystalVariables]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
  [./T]
    order = CONSTANT
    family = MONOMIAL
    [./InitialCondition]
      type = FunctionIC
      variable = T
      function = f_T
    [../]
  [../]
[]

[ICs]
  [./phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '190 490 190 490'
    y_positions = '190 190 490 490'
    z_positions = '  0   0   0   0'
    radii = '150 150 150 150'
    invalue = 0
    outvalue = 1
  [../]
  [./gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 190
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 490
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr2_IC]
    type = SmoothCircleIC
    variable = gr2
    x1 = 190
    y1 = 490
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr3_IC]
    type = SmoothCircleIC
    variable = gr3
    x1 = 490
    y1 = 490
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
[]

[Functions]
  [./f_T]
    type = ConstantFunction
    value = 1600
  [../]
[]

[Materials]
  # Free energy coefficients for parabolic curves
  [./ks]
    type = ParsedMaterial
    property_name = ks
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.0025 157.16'
    expression = 'a*T + b'
  [../]
  [./kv]
    type = ParsedMaterial
    property_name = kv
    material_property_names = 'ks'
    expression = '10*ks'
  [../]
  # Diffusivity and mobilities
  [./chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 19.7
    c = phi
    T = T
    D0 = 2.0e11
    GBmob0 = 1.4759e9
    Q = 2.77
    Em = 2.40
    bulkindex = 1
    gbindex = 20
    surfindex = 100
    outputs = exodus
  [../]
  # Equilibrium vacancy concentration
  [./cs_eq]
    type = DerivativeParsedMaterial
    property_name = cs_eq
    coupled_variables = 'gr0 gr1 gr2 gr3 T'
    constant_names = 'Ef c_GB kB'
    constant_expressions = '2.69 0.189 8.617343e-5'
    expression = 'bnds:=gr0^2 + gr1^2 + gr2^2 + gr3^2; exp(-Ef/kB/T) + 4.0 * c_GB * (1 - bnds)^2'
  [../]
  # Everything else
  [./sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 19.7
    grainboundary_energy = 9.86
    void_energy_coefficient = kv
    solid_energy_coefficient = ks
    equilibrium_vacancy_concentration = cs_eq
    solid_energy_model = PARABOLIC
  [../]
[]

[Kernels]
  [./dt_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./dt_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./dt_gr2]
    type = TimeDerivative
    variable = gr2
  [../]
  [./dt_gr3]
    type = TimeDerivative
    variable = gr3
  [../]
  [./dt_phi]
    type = TimeDerivative
    variable = phi
  [../]
  [./dt_w]
    type = TimeDerivative
    variable = w
  [../]
[]

[AuxKernels]
  [./bnds_aux]
    type = BndsCalcAux
    variable = bnds
    execute_on = 'initial timestep_end'
  [../]
  [./T_aux]
    type = FunctionAux
    variable = T
    function = f_T
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = JFNK
  dt = 1
  num_steps = 1
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialSintering_test.i)

#input file to test the materials GrandPotentialTensorMaterial

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 17
  ny = 17
  xmin = 0
  xmax = 680
  ymin = 0
  ymax = 680
  uniform_refine = 1
[]

[GlobalParams]
  op_num = 4
  var_name_base = gr
  int_width = 40
[]

[Variables]
  [./w]
  [../]
  [./phi]
  [../]
  [./PolycrystalVariables]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
  [./T]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./F_loc]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[ICs]
  [./phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '190 490 190 490'
    y_positions = '190 190 490 490'
    z_positions = '  0   0   0   0'
    radii = '150 150 150 150'
    invalue = 0
    outvalue = 1
  [../]
  [./gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 190
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 490
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr2_IC]
    type = SmoothCircleIC
    variable = gr2
    x1 = 190
    y1 = 490
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr3_IC]
    type = SmoothCircleIC
    variable = gr3
    x1 = 490
    y1 = 490
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
[]

[Functions]
  [./f_T]
    type = ConstantFunction
    value = 1600
  [../]
[]

[Materials]
  # Free energy coefficients for parabolic curves
  [./ks]
    type = ParsedMaterial
    property_name = ks
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.0025 157.16'
    expression = 'a*T + b'
  [../]
  [./kv]
    type = ParsedMaterial
    property_name = kv
    material_property_names = 'ks'
    expression = '10*ks'
  [../]
  # Diffusivity and mobilities
  [./chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 19.7
    c = phi
    T = T
    D0 = 2.0e11
    GBmob0 = 1.4759e9
    Q = 2.77
    Em = 2.40
    bulkindex = 1
    gbindex = 20
    surfindex = 100
    outputs = exodus
  [../]
  # Equilibrium vacancy concentration
  [./cs_eq]
    type = DerivativeParsedMaterial
    property_name = cs_eq
    coupled_variables = 'gr0 gr1 gr2 gr3 T'
    constant_names = 'Ef c_GB kB'
    constant_expressions = '2.69 0.189 8.617343e-5'
    expression = 'bnds:=gr0^2 + gr1^2 + gr2^2 + gr3^2; exp(-Ef/kB/T) + 4.0 * c_GB * (1 - bnds)^2'
  [../]
  # Everything else
  [./sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 19.7
    grainboundary_energy = 9.86
    void_energy_coefficient = kv
    solid_energy_coefficient = ks
    equilibrium_vacancy_concentration = cs_eq
    solid_energy_model = PARABOLIC
  [../]

  # Concentration is only meant for output
  [./c]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'hs rhos hv rhov'
    constant_names = 'Va'
    constant_expressions = '0.04092'
    expression = 'Va*(hs*rhos + hv*rhov)'
    outputs = exodus
  [../]
  [./f_bulk]
    type = ParsedMaterial
    property_name = f_bulk
    coupled_variables = 'phi gr0 gr1 gr2 gr3'
    material_property_names = 'mu gamma'
    expression = 'mu*(phi^4/4-phi^2/2 + gr0^4/4-gr0^2/2 + gr1^4/4-gr1^2/2
                  + gr2^4/4-gr2^2/2 + gr3^4/4-gr3^2/2
                  + gamma*(phi^2*(gr0^2+gr1^2+gr2^2+gr3^2) + gr0^2*(gr1^2+gr2^2+gr3^2)
                  + gr1^2*(gr2^2 + gr3^2) + gr2^2*gr3^2) + 0.25)'
    outputs = exodus
  [../]
  [./f_switch]
    type = ParsedMaterial
    property_name = f_switch
    coupled_variables = 'w'
    material_property_names = 'chi'
    expression = '0.5*w^2*chi'
    outputs = exodus
  [../]
  [./f0]
    type = ParsedMaterial
    property_name = f0
    material_property_names = 'f_bulk f_switch'
    expression = 'f_bulk + f_switch'
  [../]
[]

[Kernels]
  [./dt_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./dt_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./dt_gr2]
    type = TimeDerivative
    variable = gr2
  [../]
  [./dt_gr3]
    type = TimeDerivative
    variable = gr3
  [../]
  [./dt_phi]
    type = TimeDerivative
    variable = phi
  [../]
  [./dt_w]
    type = TimeDerivative
    variable = w
  [../]
[]

[AuxKernels]
  [./bnds_aux]
    type = BndsCalcAux
    variable = bnds
    execute_on = 'initial timestep_end'
  [../]
  [./T_aux]
    type = FunctionAux
    variable = T
    function = f_T
  [../]
  [./F_aux]
    type = TotalFreeEnergy
    variable = F_loc
    f_name = f0
    interfacial_vars = 'phi gr0 gr1 gr2 gr3'
    kappa_names = 'kappa kappa kappa kappa kappa'
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = JFNK
  dt = 1
  num_steps = 1
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/MaskedBodyForce/MaskedBodyForce_test.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 20
  ny = 20
  elem_type = QUAD
[]

[Variables]
  [./u]
  [../]
[]

[AuxVariables]
  [./c]
  [../]
[]

[ICs]
  [./initial]
    value = 1.0
    variable = u
    type = ConstantIC
  [../]
  [./c_IC]
    int_width = 0.1
    x1 = 0.5
    y1 = 0.5
    radius = 0.25
    outvalue = 0
    variable = c
    invalue = 1
    type = SmoothCircleIC
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
  [../]
  [./time]
    type = TimeDerivative
    variable = u
  [../]
  [./source]
    type = MaskedBodyForce
    variable = u
    value = 1
    mask = mask
  [../]
[]

[Materials]
  [./mask]
    type = ParsedMaterial
    expression = if(c>0.5,0,1)
    property_name = mask
    coupled_variables = c
  [../]
[]

[Executioner]
  type = Transient
  num_steps = 1
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/ad_coupled_gradient_dot/diffusion_rate.i)

# Solves the problem
# -mu * Lap(u_dot) + u_dot = alpha * Lap(u) - 2*u*(1-3*u+2*u^2)
# for mu = 1 and alpha = 0.01
# (see appendix B of A. Guevel et al. "Viscous phase-field modeling for chemo-mechanical microstructural evolution: application to geomaterials and pressure solution." In print.)

n_elem = 100
alpha = 0.01
mu = 1

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 1
  nx = '${n_elem}'
  elem_type = EDGE2
[]

[Variables]
  [u]
    [InitialCondition]
      type = ConstantIC
      value = 0.51
    []
  []
[]

[Kernels]
  [Lap]
    type = ADMatDiffusion
    variable = u
    diffusivity = '${alpha}'
  []
  [LapDot]
    type = ADDiffusionRate
    variable = u
    mu = '${mu}'
  []
  [Reac]
    type = ADMatReaction
    variable = u
    reaction_rate = L
  []
  [Visc]
    type = ADTimeDerivative
    variable = u
  []
[]

[Materials]
  [parsed]
    type = ADParsedMaterial
    expression = '-2*(1-3*u+2*u*u)'
    coupled_variables = 'u'
    property_name = 'L'
  []
[]

[BCs]
  [both]
    type = ADDirichletBC
    variable = u
    value = 0.51
    boundary = 'left right'
  []
[]

[Postprocessors]
  [mid_u]
    type = PointValue
    variable = u
    point = '0.5 0 0'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'

  num_steps = 1000
  dt = 0.1

  nl_abs_tol = 1e-9

[]

[Outputs]
  print_linear_residuals = false
  [csv]
    type = CSV
    file_base = 'diffusion_rate'
  []
[]

(modules/heat_transfer/test/tests/code_verification/cylindrical_test_no2.i)

# Problem II.2
#
# The thermal conductivity of an infinitely long hollow tube varies
# linearly with temperature. It is exposed on the inner
# and outer surfaces to constant temperatures.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    xmin = 0.2
    nx = 4
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Problem]
  coord_type = RZ
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'ri ro beta ki ko ui uo'
    symbol_values = '0.2 1.0 1e-3 5.3 5 300 0'
    expression = 'uo+(ko/beta)* ( ( 1 + beta*(ki+ko)*(ui-uo)*( log(x/ro) / log(ri/ro) )/(ko^2))^0.5 -1 )'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
[]

[BCs]
  [./ui]
    type = DirichletBC
    boundary = left
    variable = u
    value = 300
  [../]
  [./uo]
    type = DirichletBC
    boundary = right
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '5 + 1e-3 * (u-0)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(tutorials/tutorial02_multiapps/step03_coupling/02_parent_picard.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
  []
[]

[AuxVariables]
  [vt]
  []
[]

[Kernels]
  [diff]
    type = MatDiffusion
    variable = u
  []
  [force]
    type = BodyForce
    variable = u
    value = 1.
  []
  [td]
    type = TimeDerivative
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Materials]
  [diff]
    type = ParsedMaterial
    property_name = D
    coupled_variables = 'vt'
    expression = 'vt'
  []
[]

[Executioner]
  type = Transient
  end_time = 2
  dt = 0.2

  fixed_point_max_its = 10

  nl_abs_tol = 1e-10
  fixed_point_rel_tol = 1e-6
  fixed_point_abs_tol = 1e-10

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

[MultiApps]
  [micro]
    type = TransientMultiApp
    positions = '0.15 0.15 0  0.45 0.45 0  0.75 0.75 0'
    input_files = '02_sub_picard.i'
    execute_on = timestep_end
    output_in_position = true
  []
[]

[Transfers]
  [push_u]
    type = MultiAppVariableValueSampleTransfer
    to_multi_app = micro
    source_variable = u
    variable = ut
  []

  [pull_v]
    type = MultiAppPostprocessorInterpolationTransfer
    from_multi_app = micro
    variable = vt
    postprocessor = average_v
  []
[]

(modules/phase_field/examples/anisotropic_interfaces/GrandPotentialSolidification.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 28
  ny = 28
  xmin = -7
  xmax = 7
  ymin = -7
  ymax = 7
  uniform_refine = 2
[]

[GlobalParams]
  radius = 0.2
  int_width = 0.1
  x1 = 0.0
  y1 = 0.0
  derivative_order = 2
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
  [./T]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
[]

[AuxKernels]
  [./bnds]
    type = BndsCalcAux
    variable = bnds
    v = 'etaa0 etab0'
  [../]
[]

[ICs]
  [./w]
    type = SmoothCircleIC
    variable = w
    # note w = A*(c-cleq), A = 1.0, cleq = 0.0 ,i.e., w = c (in the matrix/liquid phase)
    outvalue = -4.0
    invalue = 0.0
  [../]
  [./etaa0]
    type = SmoothCircleIC
    variable = etaa0
    #Solid phase
    outvalue = 0.0
    invalue = 1.0
  [../]
  [./etab0]
    type = SmoothCircleIC
    variable = etab0
    #Liquid phase
    outvalue = 1.0
    invalue = 0.0
  [../]
[]

[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0'
    gamma_names = 'gab'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etab0 w T'
  [../]
  [./ACa0_int1]
    type = ACInterface2DMultiPhase1
    variable = etaa0
    etas = 'etab0'
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
  [../]
  [./ACa0_int2]
    type = ACInterface2DMultiPhase2
    variable = etaa0
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0'
    gamma_names = 'gab'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etaa0 w T'
  [../]
  [./ACb0_int1]
    type = ACInterface2DMultiPhase1
    variable = etab0
    etas = 'etaa0'
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
  [../]
  [./ACb0_int2]
    type = ACInterface2DMultiPhase2
    variable = etab0
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./T_dot]
    type = TimeDerivative
    variable = T
  [../]
  [./CoefDiffusion]
    type = Diffusion
    variable = T
  [../]
  [./etaa0_dot_T]
    type = CoefCoupledTimeDerivative
    variable = T
    v = etaa0
    coef = -5.0
  [../]
[]

[Materials]
  [./ha]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0'
    phase_etas = 'etaa0'
  [../]
  [./hb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0'
    phase_etas = 'etab0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w T'
    property_name = omegab
    material_property_names = 'Vm kb cbeq S Tm'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq-S*(T-Tm)'
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
  [../]
  [./kappaa]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
    etaa = etaa0
    etab = etab0
    anisotropy_strength = 0.05
    kappa_bar = 0.05
    outputs = exodus
    output_properties = 'kappaa'
  [../]
  [./kappab]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
    etaa = etab0
    etab = etaa0
    anisotropy_strength = 0.05
    kappa_bar = 0.05
    outputs = exodus
    output_properties = 'kappab'
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'L     D    chi  Vm   ka    caeq kb    cbeq  gab mu   S   Tm'
    prop_values = '33.33 1.0  0.1  1.0  10.0  0.1  10.0  0.9   4.5 10.0 1.0 5.0'
  [../]
  [./Mobility]
    type = ParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart'
  petsc_options_value = 'hypre    boomeramg      31'
   l_tol = 1.0e-3
  l_max_its = 30
  nl_max_its = 15
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 1e-10
  end_time = 2.0
  dtmax = 0.05
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.0005
    cutback_factor = 0.7
    growth_factor = 1.2
  [../]
[]

[Adaptivity]
 initial_steps = 5
 max_h_level = 3
 initial_marker = err_eta
 marker = err_bnds
[./Markers]
   [./err_eta]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_eta
   [../]
   [./err_bnds]
     type = ErrorFractionMarker
     coarsen = 0.3
     refine = 0.95
     indicator = ind_bnds
   [../]
 [../]
 [./Indicators]
   [./ind_eta]
     type = GradientJumpIndicator
     variable = etaa0
    [../]
    [./ind_bnds]
      type = GradientJumpIndicator
      variable = bnds
   [../]
 [../]
[]

[Outputs]
  time_step_interval = 5
  exodus = true
[]

(test/tests/materials/derivative_material_interface/ad_parsed_material.i)

#
# Test the parsed function free enery Allen-Cahn Bulk kernel
#

[Mesh]
  type = GeneratedMesh
  dim = 2
[]

[Variables]
  [./eta]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      function = x
    [../]
  [../]
[]

[Kernels]
  [./diff]
    type = ADMatDiffusion
    variable = eta
    diffusivity = F
  [../]
  [./dt]
    type = TimeDerivative
    variable = eta
  [../]
[]

[Materials]
  [./consts]
    type = ADParsedMaterial
    coupled_variables  = 'eta'
    expression ='(eta-0.5)^2'
    outputs = exodus
  [../]
[]

[Executioner]
  type = Transient
  num_steps = 1
  dt = 0.1
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(test/tests/materials/derivative_material_interface/ad_derivative_parsed_material.i)

#
# Test the AD version of derivative parsed material
#

[Mesh]
  type = GeneratedMesh
  dim = 2
[]

[Variables]
  [./eta]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = FunctionIC
      function = x
    [../]
  [../]
[]

[Kernels]
  [./diff]
    type = ADMatDiffusion
    variable = eta
    diffusivity = F
  []
  [./dt]
    type = TimeDerivative
    variable = eta
  []
[]

[Materials]
  [./Fbar]
    type = ADDerivativeParsedMaterial
    coupled_variables  = 'eta'
    property_name = Fbar
    expression ='1/3*(eta-0.5)^3'
  []
  [./F]
    type = ADParsedMaterial
    coupled_variables  = 'eta'
    material_property_names = 'F:=D[Fbar,eta]'
    expression ='F'
    outputs = exodus
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
  dt = 0.1
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/heat_transfer/test/tests/code_verification/cartesian_test_no2.i)

# Problem I.2
#
# An infinite plate with a thermal conductivity that varies linearly with
# temperature. Each boundary is exposed to a constant temperature.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    nx = 1
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'L beta ki ko ui uo'
    symbol_values = '1 1e-3 5.3 5 300 0'
    expression = 'uo+(ko/beta)* ( (1 + L*beta*(ki+ko)*(ui-uo)*((L-x)/(ko*L)^2) )^0.5  - 1)'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
[]

[BCs]
  [./ui]
    type = DirichletBC
    boundary = left
    variable = u
    value = 300
  [../]
  [./uo]
    type = DirichletBC
    boundary = right
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '5 + 1e-3 * (u-0)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_no_split.i)

#This input uses PhaseField-Nonconserved Action to add phase field fracture bulk rate kernels
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 10
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Modules]
  [./PhaseField]
    [./Nonconserved]
      [./c]
        free_energy = F
        kappa = kappa_op
        mobility = L
      [../]
    [../]
  [../]
  [./TensorMechanics]
    [./Master]
      [./mech]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'stress_yy'
        save_in = 'resid_x resid_y'
      [../]
    [../]
  [../]
[]

[AuxVariables]
  [./resid_x]
  [../]
  [./resid_y]
  [../]
[]

[Kernels]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
    type = DirichletBC
    variable = disp_x
    boundary = top
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.04 1e-4'
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '120.0 80.0'
    fill_method = symmetric_isotropic
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = none
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '0.0'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./resid_x]
    type = NodalSum
    variable = resid_x
    boundary = 2
  [../]
  [./resid_y]
    type = NodalSum
    variable = resid_y
    boundary = 2
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly       lu           1'

  nl_rel_tol = 1e-8
  l_max_its = 10
  nl_max_its = 10

  dt = 1e-4
  dtmin = 1e-4
  num_steps = 2
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialStrictMassConservation.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 34
  ny = 34
  xmin = 0
  xmax = 340
  ymin = 0
  ymax = 340
[]

[GlobalParams]
  op_num = 2
  var_name_base = gr
  int_width = 20
[]

[Variables]
  [w]
  []
  [c]
  []
  [phi]
  []
  [PolycrystalVariables]
  []
[]

[AuxVariables]
  [bnds]
  []
  [T]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 1600
  []
[]

[ICs]
  [phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '170 170'
    y_positions = ' 70 270'
    z_positions = '  0   0'
    radii = '100 100'
    invalue = 0
    outvalue = 1
  []
  [c_IC]
    type = SpecifiedSmoothCircleIC
    variable = c
    x_positions = '170 170'
    y_positions = ' 70 270'
    z_positions = '  0   0'
    radii = '100 100'
    invalue = 0
    outvalue = 1
  []
  [gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 170
    y1 = 70
    z1 = 0
    radius = 100
    invalue = 1
    outvalue = 0
  []
  [gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 170
    y1 = 270
    z1 = 0
    radius = 100
    invalue = 1
    outvalue = 0
  []
[]

[Materials]
  # Free energy coefficients for parabolic curves
  [./ks]
    type = ParsedMaterial
    property_name = ks
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.0017 140.16'
    expression = 'a*T + b'
  [../]
  [./kv]
    type = ParsedMaterial
    property_name = kv
    material_property_names = 'ks'
    expression = '10*ks'
  [../]
  # Diffusivity and mobilities
  [chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 6.24
    c = phi
    T = T
    D0 = 0.4366e9
    GBmob0 = 1.60e12
    Q = 4.14
    Em = 4.25
    bulkindex = 1
    gbindex = 1e6
    surfindex = 1e9
  []
  # Everything else
  [cv_eq]
    type = DerivativeParsedMaterial
    property_name = cv_eq
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef c_GB kB'
    constant_expressions = '4.37 0.1 8.617343e-5'
    derivative_order = 2
    expression = 'c_B:=exp(-Ef/kB/T); bnds:=gr0^2 + gr1^2;
                c_B + 4.0 * c_GB * (1.0 - bnds)^2'
  []
  [sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 6.24
    grainboundary_energy = 5.18
    void_energy_coefficient = kv
    solid_energy_coefficient = ks
    solid_energy_model = PARABOLIC
    equilibrium_vacancy_concentration = cv_eq
  []
[]

[Modules]
  [PhaseField]
    [GrandPotential]
      switching_function_names = 'hv hs'
      anisotropic = 'true'
      chemical_potentials = 'w'
      mobilities = 'chiD'
      susceptibilities = 'chi'
      free_energies_w = 'rhov rhos'
      gamma_gr = gamma
      mobility_name_gr = L
      kappa_gr = kappa
      free_energies_gr = 'omegav omegas'
      additional_ops = 'phi'
      gamma_grxop = gamma
      mobility_name_op = Lv
      kappa_op = kappa
      free_energies_op = 'omegav omegas'
      mass_conservation = 'true'
      concentrations = 'c'
      hj_c_min = 'hv_c_min hs_c_min'
      hj_over_kVa = 'hv_over_kVa hs_over_kVa'
    []
  []
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = PJFNK
  petsc_options_iname = '-ksp_type -pc_type -sub_pc_type -pc_asm_overlap -ksp_gmres_restart -sub_ksp_type'
  petsc_options_value = 'gmres      asm      ilu          1               31                 preonly'
  nl_max_its = 30
  l_max_its = 30
  start_time = 0
  dt = 1e-4
  num_steps = 3
[]

[Outputs]
  exodus = true
[]

(modules/combined/examples/effective_properties/effective_th_cond.i)

# This example calculates the effective thermal conductivity across a microstructure
# with circular second phase precipitates. Two methods are used to calculate the effective thermal conductivity,
# the direct method that applies a temperature to one side and a heat flux to the other,
# and the AEH method.
[Mesh] #Sets mesh size to 10 microns by 10 microns
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 100
    ny = 100
    xmax = 10
    ymax = 10
  []
  [new_nodeset]
    input = gen
    type = ExtraNodesetGenerator
    coord = '5 5'
    new_boundary = 100
  []
[]

[Variables] #Adds variables needed for two ways of calculating effective thermal cond.
  [T] #Temperature used for the direct calculation
    initial_condition = 800
  []
  [Tx_AEH] #Temperature used for the x-component of the AEH solve
    initial_condition = 800
    scaling = 1.0e4 #Scales residual to improve convergence
  []
  [Ty_AEH] #Temperature used for the y-component of the AEH solve
    initial_condition = 800
    scaling = 1.0e4  #Scales residual to improve convergence
  []
[]

[AuxVariables] #Creates second constant phase
  [phase2]
  []
[]

[ICs] #Sets the IC for the second constant phase
  [phase2_IC] #Creates circles with smooth interfaces at random locations
    variable = phase2
    type = MultiSmoothCircleIC
    int_width = 0.3
    numbub = 20
    bubspac = 1.5
    radius = 0.5
    outvalue = 0
    invalue = 1
    block = 0
  []
[]

[Kernels]
  [HtCond] #Kernel for direct calculation of thermal cond
    type = HeatConduction
    variable = T
  []
  [heat_x] #All other kernels are for AEH approach to calculate thermal cond.
    type = HeatConduction
    variable = Tx_AEH
  []
  [heat_rhs_x]
    type = HomogenizedHeatConduction
    variable = Tx_AEH
    component = 0
  []
  [heat_y]
    type = HeatConduction
    variable = Ty_AEH
  []
  [heat_rhs_y]
    type = HomogenizedHeatConduction
    variable = Ty_AEH
    component = 1
  []
[]

[BCs]
  [Periodic]
    [all]
      auto_direction = 'x y'
      variable = 'Tx_AEH Ty_AEH'
    []
  []
  [left_T] #Fix temperature on the left side
    type = DirichletBC
    variable = T
    boundary = left
    value = 800
  []
  [right_flux] #Set heat flux on the right side
    type = NeumannBC
    variable = T
    boundary = right
    value = 5e-6
  []
  [fix_x] #Fix Tx_AEH at a single point
    type = DirichletBC
    variable = Tx_AEH
    value = 800
    boundary = 100
  []
  [fix_y] #Fix Ty_AEH at a single point
    type = DirichletBC
    variable = Ty_AEH
    value = 800
    boundary = 100
  []
[]

[Materials]
  [thcond] #The equation defining the thermal conductivity is defined here, using two ifs
    # The k in the bulk is k_b, in the precipitate k_p2, and across the interaface k_int
    type = ParsedMaterial
    block = 0
    constant_names = 'length_scale k_b k_p2 k_int'
    constant_expressions = '1e-6 5 1 0.1'
    expression = 'sk_b:= length_scale*k_b; sk_p2:= length_scale*k_p2; sk_int:= k_int*length_scale; if(phase2>0.1,if(phase2>0.95,sk_p2,sk_int),sk_b)'
    outputs = exodus
    f_name = thermal_conductivity
    coupled_variables = phase2
  []
[]

[Postprocessors]
  [right_T]
    type = SideAverageValue
    variable = T
    boundary = right
  []
  [k_x_direct] #Effective thermal conductivity from direct method
    # This value is lower than the AEH value because it is impacted by second phase
    # on the right boundary
    type = ThermalConductivity
    variable = T
    flux = 5e-6
    length_scale = 1e-06
    T_hot = 800
    dx = 10
    boundary = right
  []
  [k_x_AEH] #Effective thermal conductivity in x-direction from AEH
    type = HomogenizedThermalConductivity
    chi = 'Tx_AEH Ty_AEH'
    row = 0
    col = 0
    scale_factor = 1e6 #Scale due to length scale of problem
  []
  [k_y_AEH] #Effective thermal conductivity in x-direction from AEH
    type = HomogenizedThermalConductivity
    chi = 'Tx_AEH Ty_AEH'
    row = 1
    col = 1
    scale_factor = 1e6 #Scale due to length scale of problem
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    off_diag_row = 'Tx_AEH Ty_AEH'
    off_diag_column = 'Ty_AEH Tx_AEH'
  []
[]

[Executioner]
  type = Steady
  l_max_its = 15
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart -pc_hypre_boomeramg_strong_threshold'
  petsc_options_value = 'hypre boomeramg 31 0.7'
  l_tol = 1e-04
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
  csv = true
[]

(modules/optimization/test/tests/executioners/steady_and_adjoint/nonlinear_diffusion.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 1
    ymax = 1
    nx = 10
    ny = 10
  []
[]

[Problem]
  nl_sys_names = 'nl0 adjoint'
[]

[Variables]
  [u]
  []
  [u_adjoint]
    solver_sys = adjoint
  []
[]

[Kernels]
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = D
  []
  [src]
    type = ADBodyForce
    variable = u
    value = 1
  []

  [src_adjoint]
    type = ADBodyForce
    variable = u_adjoint
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = ADDirichletBC
    variable = u
    boundary = 'top right'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = ADParsedMaterial
    property_name = D
    expression = '0.1 + 5 * u'
    coupled_variables = 'u'
  []
[]

[Postprocessors]
  [u_avg]
    type = ElementAverageValue
    variable = u
  []
  [u_adjoint_avg]
    type = ElementAverageValue
    variable = u_adjoint
    execute_on = ADJOINT_TIMESTEP_END
  []
  [inner_product]
    type = VariableInnerProduct
    variable = u
    second_variable = u_adjoint
    execute_on = ADJOINT_TIMESTEP_END
  []
[]

[Executioner]
  type = SteadyAndAdjoint
  forward_system = nl0
  adjoint_system = adjoint

  nl_rel_tol = 1e-12
  l_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(modules/combined/test/tests/phase_field_fracture/crack2d_aniso_cleavage_plane.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 20
    ymax = 0.5
  []
  [./noncrack]
    type = BoundingBoxNodeSetGenerator
    new_boundary = noncrack
    bottom_left = '0.5 0 0'
    top_right = '1 0 0'
    input = gen
  [../]
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [./c]
    family = LAGRANGE
    order = FIRST
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./All]
        add_variables = true
        strain = SMALL
        additional_generate_output = 'strain_yy stress_yy'
        planar_formulation = PLANE_STRAIN
      [../]
    [../]
  [../]
[]

[Kernels]
  [./ACbulk]
    type = AllenCahn
    variable = c
    f_name = F
  [../]
  [./ACInterfaceCleavageFracture]
    type = ACInterfaceCleavageFracture
    variable = c
    beta_penalty = 1
    cleavage_plane_normal = '-0.707 0.707 0.0'
  [../]
  [./dcdt]
    type = TimeDerivative
    variable = c
  [../]
  [./solid_x]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_x
    component = 0
    c = c
  [../]
  [./solid_y]
    type = PhaseFieldFractureMechanicsOffDiag
    variable = disp_y
    component = 1
    c = c
  [../]
  [./off_disp]
    type = AllenCahnElasticEnergyOffDiag
    variable = c
    displacements = 'disp_x disp_y'
    mob_name = L
  [../]
[]

[BCs]
  [./ydisp]
    type = FunctionDirichletBC
    preset = true
    variable = disp_y
    boundary = top
    function = 't'
  [../]
  [./yfix]
    type = DirichletBC
    preset = true
    variable = disp_y
    boundary = noncrack
    value = 0
  [../]
  [./xfix]
  type = DirichletBC
  preset = true
    variable = disp_x
    boundary = right
    value = 0
  [../]
[]

[Materials]
  [./pfbulkmat]
    type = GenericConstantMaterial
    prop_names = 'gc_prop l visco'
    prop_values = '1e-3 0.05 1e-6'
  [../]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '127.0 70.8 70.8 127.0 70.8 127.0 73.55 73.55 73.55'
    fill_method = symmetric9
    euler_angle_1 = 30
    euler_angle_2 = 0
    euler_angle_3 = 0
  [../]
  [./define_mobility]
    type = ParsedMaterial
    material_property_names = 'gc_prop visco'
    property_name = L
    expression = '1.0/(gc_prop * visco)'
  [../]
  [./define_kappa]
    type = ParsedMaterial
    material_property_names = 'gc_prop l'
    property_name = kappa_op
    expression = 'gc_prop * l'
  [../]
  [./damage_stress]
    type = ComputeLinearElasticPFFractureStress
    c = c
    E_name = 'elastic_energy'
    D_name = 'degradation'
    F_name = 'local_fracture_energy'
    decomposition_type = stress_spectral
  [../]
  [./degradation]
    type = DerivativeParsedMaterial
    property_name = degradation
    coupled_variables = 'c'
    expression = '(1.0-c)^2*(1.0 - eta) + eta'
    constant_names       = 'eta'
    constant_expressions = '1.0e-6'
    derivative_order = 2
  [../]
  [./local_fracture_energy]
    type = DerivativeParsedMaterial
    property_name = local_fracture_energy
    coupled_variables = 'c'
    material_property_names = 'gc_prop l'
    expression = 'c^2 * gc_prop / 2 / l'
    derivative_order = 2
  [../]
  [./fracture_driving_energy]
    type = DerivativeSumMaterial
    coupled_variables = c
    sum_materials = 'elastic_energy local_fracture_energy'
    derivative_order = 2
    property_name = F
  [../]
[]

[Postprocessors]
  [./av_stress_yy]
    type = ElementAverageValue
    variable = stress_yy
  [../]
  [./av_strain_yy]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solving_package'
  petsc_options_value = 'lu superlu_dist'

  nl_rel_tol = 1e-8
  l_tol = 1e-4
  l_max_its = 100
  nl_max_its = 10

  dt = 5e-5
  num_steps = 5
[]

[Outputs]
  exodus = true
[]

(modules/solid_mechanics/test/tests/ad_viscoplasticity_stress_update/negative_porosity.i)

# This test provides an example of an individual LPS viscoplasticity model

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmax = 0.002
  ymax = 0.002
[]

[Physics/SolidMechanics/QuasiStatic/All]
  strain = FINITE
  add_variables = true
  generate_output = 'strain_xx strain_yy strain_xy hydrostatic_stress vonmises_stress'
  use_automatic_differentiation = true
[]

[Functions]
  [./pull]
    type = PiecewiseLinear
    x = '0 0.1'
    y = '0 1e-5'
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1e10
    poissons_ratio = 0.3
  [../]
  [./stress]
    type = ADComputeMultipleInelasticStress
    inelastic_models = lps
    outputs = all
  [../]
  [./porosity]
    type = ADGenericConstantMaterial
    prop_names = 'porosity'
    prop_values = '-0.1'
    outputs = 'all'
  [../]

  [./lps]
    type = ADViscoplasticityStressUpdate
    coefficient = 'coef'
    power = 3
    outputs = all
    relative_tolerance = 1e-11
    initial_porosity = 0.1
    negative_behavior = ZERO
  [../]
  [./coef]
    type = ADParsedMaterial
    property_name = coef
    # Example of creep power law
    expression = '1e-18 * exp(-4e4 / 1.987 / 1200)'
  [../]
[]

[BCs]
  [./no_disp_x]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./no_disp_y]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./pull_disp_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = top
    function = pull
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 0.01
  end_time = 0.12
[]

[Postprocessors]
  [./disp_x]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  [../]
  [./disp_y]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
  [./avg_hydro]
    type = ElementAverageValue
    variable = hydrostatic_stress
  [../]
  [./avg_vonmises]
    type = ElementAverageValue
    variable = vonmises_stress
  [../]
  [./dt]
    type = TimestepSize
  [../]
  [./num_lin]
    type = NumLinearIterations
    outputs = console
  [../]
  [./num_nonlin]
    type = NumNonlinearIterations
    outputs = console
  [../]
  [./eff_creep_strain]
    type = ElementAverageValue
    variable = effective_viscoplasticity
  [../]
  [./porosity]
    type = ElementAverageValue
    variable = porosity
  [../]
[]

[Outputs]
  csv = true
[]

(modules/phase_field/test/tests/MultiPhase/penalty.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 14
  ny = 10
  nz = 0
  xmin = 10
  xmax = 40
  ymin = 15
  ymax = 35
  elem_type = QUAD4
[]

[GlobalParams]
  penalty = 5
[]

[Variables]
  [./c]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = 25.0
      y1 = 25.0
      radius = 6.0
      invalue = 0.9
      outvalue = 0.1
      int_width = 3.0
    [../]
  [../]
  [./w]
    order = FIRST
    family = LAGRANGE
  [../]

  [./eta1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = SmoothCircleIC
      x1 = 30.0
      y1 = 25.0
      radius = 4.0
      invalue = 0.9
      outvalue = 0.1
      int_width = 2.0
    [../]
  [../]
  [./eta2]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  [../]
[]

[Kernels]
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta1
    coupled_variables = 'c eta2'
    f_name = F
  [../]
  [./ACInterface1]
    type = ACInterface
    variable = eta1
    kappa_name = kappa_eta
  [../]
  [./penalty1]
    type = SwitchingFunctionPenalty
    variable = eta1
    etas    = 'eta1 eta2'
    h_names = 'h1   h2'
  [../]

  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulk2]
    type = AllenCahn
    variable = eta2
    coupled_variables = 'c eta1'
    f_name = F
  [../]
  [./ACInterface2]
    type = ACInterface
    variable = eta2
    kappa_name = kappa_eta
  [../]
  [./penalty2]
    type = SwitchingFunctionPenalty
    variable = eta2
    etas    = 'eta1 eta2'
    h_names = 'h1   h2'
  [../]

  [./c_res]
    type = SplitCHParsed
    variable = c
    f_name = F
    kappa_name = kappa_c
    w = w
    coupled_variables = 'eta1 eta2'
  [../]
  [./w_res]
    type = SplitCHWRes
    variable = w
    mob_name = M
  [../]
  [./time1]
    type = CoupledTimeDerivative
    variable = w
    v = c
  [../]
[]

[BCs]
  [./Periodic]
    [./All]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Materials]
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'L kappa_eta'
    prop_values = '1 1        '
  [../]
  [./consts2]
    type = GenericConstantMaterial
    prop_names  = 'M kappa_c'
    prop_values = '1 1'
  [../]

  [./hsum]
    type = ParsedMaterial
    expression = h1+h2
    property_name = hsum
    material_property_names = 'h1 h2'
    coupled_variables = 'c'
    outputs = exodus
  [../]

  [./switching1]
    type = SwitchingFunctionMaterial
    function_name = h1
    eta = eta1
    h_order = SIMPLE
  [../]
  [./switching2]
    type = SwitchingFunctionMaterial
    function_name = h2
    eta = eta2
    h_order = SIMPLE
  [../]

  [./barrier]
    type = MultiBarrierFunctionMaterial
    etas = 'eta1 eta2'
  [../]

  [./free_energy_A]
    type = DerivativeParsedMaterial
    property_name = Fa
    coupled_variables = 'c'
    expression = '(c-0.1)^2'
    derivative_order = 2
  [../]
  [./free_energy_B]
    type = DerivativeParsedMaterial
    property_name = Fb
    coupled_variables = 'c'
    expression = '(c-0.9)^2'
    derivative_order = 2
  [../]

  [./free_energy]
    type = DerivativeMultiPhaseMaterial
    property_name = F
    fi_names = 'Fa   Fb'
    hi_names = 'h1   h2'
    etas     = 'eta1 eta2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = 'bdf2'

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm       lu'

  l_max_its = 15
  l_tol = 1.0e-6

  nl_max_its = 50
  nl_rel_tol = 1.0e-7
  nl_abs_tol = 1.0e-9

  start_time = 0.0
  num_steps = 2
  dt = 0.05
  dtmin = 0.01
[]

[Debug]
  # show_var_residual_norms = true
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/phase_field/test/tests/SoretDiffusion/direct_temp.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 30
  xmax = 500
  elem_type = EDGE
[]

[GlobalParams]
  polynomial_order = 8
[]

[Variables]
  [./c]
    family = HERMITE
    order = THIRD
  [../]
  [./T]
    initial_condition = 1000.0
    scaling = 1.0e5
  [../]
[]

[ICs]
  [./c_IC]
    type = SmoothCircleIC
    x1 = 125.0
    y1 = 0.0
    radius = 60.0
    invalue = 1.0
    outvalue = 0.1
    int_width = 100.0
    variable = c
  [../]
[]

[Kernels]
  [./c_int]
    type = CHInterface
    variable = c
    kappa_name = kappa
    mob_name = M
  [../]
  [./c_bulk]
    type = CahnHilliard
    variable = c
    mob_name = M
    f_name = F
  [../]
  [./c_soret]
    type = SoretDiffusion
    variable = c
    T = T
    diff_name = D
    Q_name = Qstar
  [../]
  [./c_dot]
    type = TimeDerivative
    variable = c
  [../]
  [./HtCond]
    type = MatDiffusion
    variable = T
    diffusivity = thermal_conductivity
  [../]
[]

[BCs]
  [./Left_T]
    type = DirichletBC
    variable = T
    boundary = left
    value = 1000.0
  [../]

  [./Right_T]
    type = DirichletBC
    variable = T
    boundary = right
    value = 1015.0
  [../]
[]

[Materials]
  [./Copper]
    type = PFParamsPolyFreeEnergy
    c = c
    T = T # K
    int_width = 60.0
    length_scale = 1.0e-9
    time_scale = 1.0e-9
    D0 = 3.1e-5 # m^2/s, from Brown1980
    Em = 0.71 # in eV, from Balluffi1978 Table 2
    Ef = 1.28 # in eV, from Balluffi1978 Table 2
    surface_energy = 0.708 # Total guess
  [../]
  [./thcond]
    type = ParsedMaterial
    coupled_variables = 'c'
    expression = 'if(c>0.7,1e-8,4e-8)'
    property_name = thermal_conductivity
    outputs = exodus
  [../]
  [./free_energy]
    type = PolynomialFreeEnergy
    c = c
    derivative_order = 3
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = 'bdf2'
  solve_type = 'PJFNK'

  l_max_its = 30
  l_tol = 1.0e-4
  nl_max_its = 25
  nl_rel_tol = 1.0e-9

  num_steps = 60
  dt = 8.0
[]

[Outputs]
  exodus = true
[]

(modules/heat_transfer/test/tests/code_verification/cylindrical_test_no3.i)

# Problem II.3
#
# The thermal conductivity of an infinitely long hollow cylinder varies
# linearly with temperature: k = k0(1+beta*u). The tube inside radius is ri and
# outside radius is ro. It has a constant internal heat generation q and
# is exposed to the same constant temperature on both surfaces: u(ri) = u(ro) = uo.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    xmin = 0.2
    nx = 4
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Problem]
  coord_type = RZ
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'q k0 ri ro beta u0'
    symbol_values = '1200 1 0.2 1.0 1e-3 0'
    expression = 'u0+(1/beta)*( ( 1 + 0.5*beta*((ro^2-x^2)-(ro^2-ri^2) * log(ro/x)/log(ro/ri))*q/k0 )^0.5  - 1)'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
  [./heatsource]
    type = HeatSource
    function = 1200
    variable = u
  [../]
[]

[BCs]
  [./uo]
    type = DirichletBC
    boundary = 'left right'
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '1 * (1 + 1e-3*u)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/SinteringIdeal.i)

#input file to test the GrandPotentialSinteringMaterial using the ideal energy profile

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 17
  ny = 10
  xmin = 0
  xmax = 660
  ymin = 0
  ymax = 380
[]

[GlobalParams]
  op_num = 2
  var_name_base = gr
  int_width = 40
[]

[Variables]
  [./w]
    [./InitialCondition]
      type = FunctionIC
      variable = w
      function = f_w
    [../]
  [../]
  [./phi]
  [../]
  [./PolycrystalVariables]
  [../]
[]

[AuxVariables]
  [./T]
    order = CONSTANT
    family = MONOMIAL
    [./InitialCondition]
      type = FunctionIC
      variable = T
      function = f_T
    [../]
  [../]
[]

[ICs]
  [./phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '190 470'
    y_positions = '190 190'
    z_positions = '  0   0'
    radii = '150 150'
    invalue = 0
    outvalue = 1
  [../]
  [./gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 190
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 470
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
[]

[Functions]
  [./f_T]
    type = ConstantFunction
    value = 1600
  [../]
  [./f_w]
    type = ParsedFunction
    expression = '1.515e-7 * x'
  [../]
[]

[Materials]
  # Free energy coefficients for parabolic curve
  [./kv]
    type = ParsedMaterial
    property_name = kv
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.025 1571.6'
    expression = 'a*T + b'
  [../]
  # Diffusivity and mobilities
  [./chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 19.7
    c = phi
    T = T
    D0 = 2.0e11
    GBmob0 = 1.4759e9
    Q = 2.77
    Em = 2.40
    bulkindex = 1
    gbindex = 20
    surfindex = 100
  [../]
  # Equilibrium vacancy concentration
  [./cs_eq]
    type = DerivativeParsedMaterial
    property_name = cs_eq
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef Egb kB'
    constant_expressions = '2.69 2.1 8.617343e-5'
    expression = 'bnds:=gr0^2 + gr1^2; cb:=exp(-Ef/kB/T); cgb:=exp(-(Ef-Egb)/kB/T);
                cb + 4.0*(cgb-cb)*(1.0 - bnds)^2'
  [../]
  # Everything else
  [./sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 19.7
    grainboundary_energy = 9.86
    void_energy_coefficient = kv
    equilibrium_vacancy_concentration = cs_eq
    solid_energy_model = IDEAL
    outputs = exodus
  [../]

  # Concentration is only meant for output
  [./c]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'hs rhos hv rhov'
    constant_names = 'Va'
    constant_expressions = '0.04092'
    expression = 'Va*(hs*rhos + hv*rhov)'
    outputs = exodus
  [../]
[]

[Kernels]
  [./dt_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./dt_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./dt_phi]
    type = TimeDerivative
    variable = phi
  [../]
  [./dt_w]
    type = TimeDerivative
    variable = w
  [../]
[]

[AuxKernels]
  [./T_aux]
    type = FunctionAux
    variable = T
    function = f_T
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = JFNK
  dt = 1
  num_steps = 2
  nl_abs_tol = 1e-10
[]

[Outputs]
  exodus = true
[]

(modules/heat_transfer/test/tests/code_verification/spherical_test_no2.i)

# Problem III.2
#
# A spherical shell has a thermal conductivity that varies linearly
# with temperature. The inside and outside surfaces of the shell are
# exposed to constant temperatures.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    xmin = 0.2
    nx = 4
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Problem]
  coord_type = RSPHERICAL
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'ri ro beta ki ko ui uo'
    symbol_values = '0.2 1.0 1e-3 5.3 5 300 0'
    expression = 'uo+(ko/beta)* ( ( 1 + beta*(ki+ko)*(ui-uo)*( (1/x-1/ro) / (1/ri-1/ro) )/(ko^2))^0.5 -1 )'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
[]

[BCs]
  [./ui]
    type = DirichletBC
    boundary = left
    variable = u
    value = 300
  [../]
  [./uo]
    type = DirichletBC
    boundary = right
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '5 + 1e-3 * (u-0)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(test/tests/outputs/csv/csv_transient_vpp.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  parallel_type = replicated
[]

[Variables]
  [u]
  []
[]

[Materials]
  [D]
    # we need to make sure not to supply derivatives to have a
    # wrong Jacobian to force more iterations to test the output on
    type = ParsedMaterial
    property_name = D
    expression = 'u^2+0.1'
    coupled_variables = u
  []
[]

[Kernels]
  [diff]
    type = MatDiffusion
    variable = u
    diffusivity = D
  []
  [time]
    type = TimeDerivative
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[VectorPostprocessors]
  [nodes]
    type = NodalValueSampler
    boundary = top
    sort_by = x
    variable = u
    execute_on = 'INITIAL NONLINEAR LINEAR TIMESTEP_END'
  []
[]

[Executioner]
  type = Transient
  num_steps = 3
  dt = 0.1
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  verbose = true
[]

[Outputs]
  [out]
    type = CSV
    execute_on = 'LINEAR'
  []
[]

(modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialAnisotropy.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 15
  ny = 15
  xmin = -2
  xmax = 2
  ymin = -2
  ymax = 2
[]

# enable_jit set to false in many materials to make this test start up faster.
# It is recommended to set enable_jit = true or just remove these lines for
# production runs with this model
[GlobalParams]
  radius = 1.0
  int_width = 0.8
  x1 = 0
  y1 = 0
  derivative_order = 2
  enable_jit = false
[]

[Variables]
  [./w]
  [../]
  [./etaa0]
  [../]
  [./etab0]
  [../]
[]

[AuxVariables]
  [./bnds]
  [../]
[]

[AuxKernels]
  [./bnds]
    type = BndsCalcAux
    variable = bnds
    v = 'etaa0 etab0'
  [../]
[]

[ICs]
  [./w]
    type = SmoothCircleIC
    variable = w
    # note w = A*(c-cleq), A = 1.0, cleq = 0.0 ,i.e., w = c (in the matrix/liquid phase)
    outvalue = -4.0
    invalue = 0.0
  [../]
  [./etaa0]
    type = SmoothCircleIC
    variable = etaa0
    #Solid phase
    outvalue = 0.0
    invalue = 1.0
  [../]
  [./etab0]
    type = SmoothCircleIC
    variable = etab0
    #Liquid phase
    outvalue = 1.0
    invalue = 0.0
  [../]
[]

[BCs]
  [./Periodic]
    [./w]
      variable = w
      auto_direction = 'x y'
    [../]
    [./etaa0]
      variable = etaa0
      auto_direction = 'x y'
    [../]
    [./etab0]
      variable = etab0
      auto_direction = 'x y'
    [../]
  [../]
[]

[Kernels]
# Order parameter eta_alpha0
  [./ACa0_bulk]
    type = ACGrGrMulti
    variable = etaa0
    v =           'etab0'
    gamma_names = 'gab'
  [../]
  [./ACa0_sw]
    type = ACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etab0 w'
  [../]
  [./ACa0_int1]
    type = ACInterface2DMultiPhase1
    variable = etaa0
    etas = 'etab0'
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
  [../]
  [./ACa0_int2]
    type = ACInterface2DMultiPhase2
    variable = etaa0
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
  [../]
  [./ea0_dot]
    type = TimeDerivative
    variable = etaa0
  [../]
# Order parameter eta_beta0
  [./ACb0_bulk]
    type = ACGrGrMulti
    variable = etab0
    v =           'etaa0'
    gamma_names = 'gab'
  [../]
  [./ACb0_sw]
    type = ACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab'
    hj_names  = 'ha     hb'
    coupled_variables = 'etaa0 w'
  [../]
  [./ACb0_int1]
    type = ACInterface2DMultiPhase1
    variable = etab0
    etas = 'etaa0'
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
  [../]
  [./ACb0_int2]
    type = ACInterface2DMultiPhase2
    variable = etab0
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
  [../]
  [./eb0_dot]
    type = TimeDerivative
    variable = etab0
  [../]
#Chemical potential
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = Dchi
  [../]
  [./coupled_etaa0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
  [./coupled_etab0dot]
    type = CoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob'
    hj_names = 'ha   hb'
    coupled_variables = 'etaa0 etab0'
  [../]
[]

[Materials]
  [./ha]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0'
    phase_etas = 'etaa0'
  [../]
  [./hb]
    type = SwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0'
    phase_etas = 'etab0'
  [../]
  [./omegaa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegaa
    material_property_names = 'Vm ka caeq'
    expression = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
  [../]
  [./omegab]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = omegab
    material_property_names = 'Vm kb cbeq'
    expression = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq'
  [../]
  [./rhoa]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhoa
    material_property_names = 'Vm ka caeq'
    expression = 'w/Vm^2/ka + caeq/Vm'
  [../]
  [./rhob]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = rhob
    material_property_names = 'Vm kb cbeq'
    expression = 'w/Vm^2/kb + cbeq/Vm'
  [../]
  [./kappaa]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappaa
    dkappadgrad_etaa_name = dkappadgrad_etaa
    d2kappadgrad_etaa_name = d2kappadgrad_etaa
    etaa = etaa0
    etab = etab0
    outputs = exodus
    output_properties = 'kappaa dkappadgrad_etaa'
  [../]
  [./kappab]
    type = InterfaceOrientationMultiphaseMaterial
    kappa_name = kappab
    dkappadgrad_etaa_name = dkappadgrad_etab
    d2kappadgrad_etaa_name = d2kappadgrad_etab
    etaa = etab0
    etab = etaa0
    outputs = exodus
    output_properties = 'kappab dkappadgrad_etab'
  [../]
  [./const]
    type = GenericConstantMaterial
    prop_names =  'L   D    chi  Vm   ka    caeq kb    cbeq  gab mu'
    prop_values = '1.0 1.0  0.1  1.0  10.0  0.1  10.0  0.9   4.5 10.0'
  [../]
  [./Mobility]
    type = ParsedMaterial
    property_name = Dchi
    material_property_names = 'D chi'
    expression = 'D*chi'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  l_tol = 1.0e-5
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1e-12
  num_steps = 2
  dt = 0.001
[]

[Outputs]
  exodus = true
[]

(test/tests/bcs/mat_neumann_bc/mat_neumann.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 10
  ymax = 10
  nx = 10
  ny = 10
[]

[Variables]
  [./u]
  [../]
[]

[AuxVariables]
  [./phi]
  [../]
[]

[ICs]
  [./phi_IC]
    type = FunctionIC
    variable = phi
    function = ic_func_phi
  [../]
[]

[Functions]
  [./ic_func_phi]
    type = ParsedFunction
    expression = '0.5 * (1 - tanh((x - 5) / 0.8))'
  [../]
[]

[BCs]
  [./top]
    type = MatNeumannBC
    variable = u
    boundary = top
    value = 2
    boundary_material = hm
  [../]
[]

[Kernels]
  [./dudt]
    type = TimeDerivative
    variable = u
  [../]
  [./diff]
    type = Diffusion
    variable = u
  [../]
[]

[Materials]
  [./hm]
    type = ParsedMaterial
    property_name = hm
    coupled_variables = 'phi'
    expression = '3*phi^2 - 2*phi^3'
    outputs = exodus
  [../]
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'
  end_time = 10
[]

[Outputs]
  exodus = true
[]

(modules/heat_transfer/test/tests/code_verification/spherical_test_no3.i)

# Problem III.3
#
# The thermal conductivity of a spherical shell varies linearly with
# temperature: k = k0(1+beta* u). The inside radius is ri and the outside radius
# is ro. It has a constant internal heat generation q and is exposed to
# the same constant temperature on both surfaces: u(ri) = u(ro) = uo.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    nx = 4
    xmin = 0.2
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Problem]
  coord_type = RSPHERICAL
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'q k0 ri ro beta u0'
    symbol_values = '1200 1 0.2 1.0 1e-3 0'
    expression = 'u0+(1/beta)*( ( 1 + (1/3)*beta*((ro^2-x^2)-(ro^2-ri^2) * (1/x-1/ro)/(1/ri-1/ro))*q/k0 )^0.5  - 1)'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
  [./heatsource]
    type = HeatSource
    function = 1200
    variable = u
  [../]
[]

[BCs]
  [./uo]
    type = DirichletBC
    boundary = 'left right'
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '1 * (1 + 1e-3*u)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(modules/combined/examples/optimization/thermomechanical/thermal_sub.i)

vol_frac = 0.4

power = 2.0

E0 = 1.0e-6
E1 = 1.0

rho0 = 0.0
rho1 = 1.0

C0 = 1.0e-6
C1 = 1.0

TC0 = 1.0e-16
TC1 = 1.0

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 40
    xmin = 0
    xmax = 40
    ymin = 0
    ymax = 40
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push_left]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push_left
    coord = '16 0 0'
  []
  [push_center]
    type = ExtraNodesetGenerator
    input = push_left
    new_boundary = push_center
    coord = '24 0 0'
  []
  [extra]
    type = SideSetsFromBoundingBoxGenerator
    input = push_center
    bottom_left = '-0.01 17.999  0'
    top_right = '5 22.001  0'
    boundary_new = n1
    boundaries_old = left
  []
  [dirichlet_bc]
    type = SideSetsFromNodeSetsGenerator
    input = extra
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 100.0
  []
[]

[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Tc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Cost]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = FIRST
    initial_condition = ${vol_frac}
  []
[]

[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [left_n1]
    type = DirichletBC
    variable = temp
    boundary = n1
    value = 0.0
  []
  [top]
    type = NeumannBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = NeumannBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = NeumannBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = NeumannBC
    variable = temp
    boundary = left
    value = 0
  []
[]

[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
[]

[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${TC0}-${TC1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${TC0}-A1*${rho0}^${power}; TC1:=A1*mat_den^${power}+B1; TC1"
    coupled_variables = 'mat_den'
    property_name = thermal_cond
    outputs = 'exodus'
  []
  [thermal_compliance]
    type = ThermalCompliance
    temperature = temp
    thermal_conductivity = thermal_cond
    outputs = 'exodus'
  []
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${E0}-${E1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${E0}-A1*${rho0}^${power}; E1:=A1*mat_den^${power}+B1; E1"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^(1/${power})-${rho1}^(1/${power})); "
                 "B1:=${C0}-A1*${rho0}^(1/${power}); C1:=A1*mat_den^(1/${power})+B1; C1"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
  []
  [CostDensity]
    type = ParsedMaterial
    property_name = CostDensity
    coupled_variables = 'mat_den Cost'
    expression = 'mat_den*Cost'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = Cost_mat
    outputs = 'exodus'
  []
  [tc]
    type = ThermalSensitivity
    design_density = mat_den
    thermal_conductivity = thermal_cond
    temperature = temp
    outputs = 'exodus'
  []
[]

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

[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = cost_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_thermal]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Cc
  [calc_sense_cost]
    type = SensitivityFilter
    density_sensitivity = Cc
    design_density = mat_den
    filter_UO = rad_avg_cost
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Tc
  [calc_sense_thermal]
    type = SensitivityFilter
    density_sensitivity = Tc
    design_density = mat_den
    filter_UO = rad_avg_thermal
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-12
  dt = 1.0
  num_steps = 500
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [right_flux]
    type = SideDiffusiveFluxAverage
    variable = temp
    boundary = right
    diffusivity = 10
  []
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralMaterialProperty
    mat_prop = CostDensity
  []
  [cost_frac]
    type = ParsedPostprocessor
    expression = 'cost / mesh_volume'
    pp_names = 'cost mesh_volume'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [objective_thermal]
    type = ElementIntegralMaterialProperty
    mat_prop = thermal_compliance
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

(modules/heat_transfer/test/tests/code_verification/cartesian_test_no3.i)

# Problem I.3
#
# The thermal conductivity of an infinite plate varies linearly with
# temperature: k = ko(1+beta*u). It has a constant internal heat generation q,
# and has the boundary conditions du/dx = 0 at x= L and u(L) = uo.
#
# REFERENCE:
# A. Toptan, et al. (Mar.2020). Tech. rep. CASL-U-2020-1939-000, SAND2020-3887 R. DOI:10.2172/1614683.

[Mesh]
  [./geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type = EDGE2
    nx = 4
  [../]
[]

[Variables]
  [./u]
    order = FIRST
  [../]
[]

[Functions]
  [./exact]
    type = ParsedFunction
    symbol_names = 'q L beta uo ko'
    symbol_values = '1200 1 1e-3 0 1'
    expression = 'uo+(1/beta)*( ( 1 + (1-(x/L)^2) * (beta*q*L^2) / ko )^0.5  - 1)'
  [../]
[]

[Kernels]
  [./heat]
    type = HeatConduction
    variable = u
  [../]
  [./heatsource]
    type = HeatSource
    function = 1200
    variable = u
  [../]
[]

[BCs]
  [./ui]
    type = NeumannBC
    boundary = left
    variable = u
    value = 0
  [../]
  [./uo]
    type = DirichletBC
    boundary = right
    variable = u
    value = 0
  [../]
[]

[Materials]
  [./property]
    type = GenericConstantMaterial
    prop_names = 'density specific_heat'
    prop_values = '1.0 1.0'
  [../]
  [./thermal_conductivity]
    type = ParsedMaterial
    property_name = 'thermal_conductivity'
    coupled_variables = u
    expression = '1 * (1 + 1e-3*u)'
  [../]
[]

[Executioner]
  type = Steady
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = exact
    variable = u
  [../]
  [./h]
    type = AverageElementSize
  []
[]

[Outputs]
  csv = true
[]

(modules/combined/test/tests/optimization/compliance_sensitivity/thermal_test.i)

vol_frac = 0.4
cost_frac = 10.0

power = 2.0

E0 = 1.0e-6
E1 = 1.0

rho0 = 0.0
rho1 = 1.0

C0 = 1.0e-6
C1 = 1.0

TC0 = 1.0e-16
TC1 = 1.0

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = 0
    xmax = 40
    ymin = 0
    ymax = 40
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push_left]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push_left
    coord = '16 0 0'
  []
  [push_center]
    type = ExtraNodesetGenerator
    input = push_left
    new_boundary = push_center
    coord = '24 0 0'
  []
  [extra]
    type = SideSetsFromBoundingBoxGenerator
    input = push_center
    bottom_left = '-0.01 17.999  0'
    top_right = '5 22.001  0'
    boundary_new = n1
    included_boundaries = left
  []
  [dirichlet_bc]
    type = SideSetsFromNodeSetsGenerator
    input = extra
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 100.0
  []
[]

[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Tc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [Cost]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = FIRST
    initial_condition = ${vol_frac}
  []
[]

[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [left_n1]
    type = DirichletBC
    variable = temp
    boundary = n1
    value = 0.0
  []
  [top]
    type = NeumannBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = NeumannBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = NeumannBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = NeumannBC
    variable = temp
    boundary = left
    value = 0
  []
[]

[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
[]

[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${TC0}-${TC1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${TC0}-A1*${rho0}^${power}; TC1:=A1*mat_den^${power}+B1; TC1"
    coupled_variables = 'mat_den'
    property_name = thermal_cond
    outputs = 'exodus'
  []
  [thermal_compliance]
    type = ThermalCompliance
    temperature = temp
    thermal_conductivity = thermal_cond
    outputs = 'exodus'
  []
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${E0}-${E1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${E0}-A1*${rho0}^${power}; E1:=A1*mat_den^${power}+B1; E1"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^(1/${power})-${rho1}^(1/${power})); "
                 "B1:=${C0}-A1*${rho0}^(1/${power}); C1:=A1*mat_den^(1/${power})+B1; C1"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
  []
  [CostDensity]
    type = ParsedMaterial
    property_name = CostDensity
    coupled_variables = 'mat_den Cost'
    expression = 'mat_den*Cost'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = Cost_mat
    outputs = 'exodus'
  []
  [tc]
    type = ThermalSensitivity
    design_density = mat_den
    thermal_conductivity = thermal_cond
    temperature = temp
    outputs = 'exodus'
  []
[]

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

[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = cost_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_thermal]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdateTwoConstraints
    density_sensitivity = Dc
    cost_density_sensitivity = Cc
    cost = Cost
    cost_fraction = ${cost_frac}
    design_density = mat_den
    volume_fraction = ${vol_frac}

    bisection_lower_bound = 0
    bisection_upper_bound = 1.0e12 # 100

    use_thermal_compliance = true
    thermal_sensitivity = Tc
    weight_mechanical_thermal = '0 1'

    relative_tolerance = 1.0e-12
    bisection_move = 0.015
    adaptive_move = false
    execute_on = TIMESTEP_BEGIN
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Cc
  [calc_sense_cost]
    type = SensitivityFilter
    density_sensitivity = Cc
    design_density = mat_den
    filter_UO = rad_avg_cost
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Tc
  [calc_sense_thermal]
    type = SensitivityFilter
    density_sensitivity = Tc
    design_density = mat_den
    filter_UO = rad_avg_thermal
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-12
  dt = 1.0
  num_steps = 5
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [right_flux]
    type = SideDiffusiveFluxAverage
    variable = temp
    boundary = right
    diffusivity = 10
  []
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralMaterialProperty
    mat_prop = CostDensity
  []
  [cost_frac]
    type = ParsedPostprocessor
    expression = 'cost / mesh_volume'
    pp_names = 'cost mesh_volume'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [objective_thermal]
    type = ElementIntegralMaterialProperty
    mat_prop = thermal_compliance
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

(test/tests/bcs/mat_neumann_bc/ad_mat_neumann.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 10
  ymax = 10
  nx = 10
  ny = 10
[]

[Variables]
  [./u]
  [../]
[]

[AuxVariables]
  [./phi]
  [../]
[]

[ICs]
  [./phi_IC]
    type = FunctionIC
    variable = phi
    function = ic_func_phi
  [../]
[]

[Functions]
  [./ic_func_phi]
    type = ParsedFunction
    expression = '0.5 * (1 - tanh((x - 5) / 0.8))'
  [../]
[]

[BCs]
  [./top]
    type = ADMatNeumannBC
    variable = u
    boundary = top
    value = 2
    boundary_material = hm
  [../]
[]

[Kernels]
  [./dudt]
    type = ADTimeDerivative
    variable = u
  [../]
  [./diff]
    type = ADDiffusion
    variable = u
  [../]
[]

[Materials]
  [./hm]
    type = ADParsedMaterial
    property_name = hm
    coupled_variables = 'phi'
    expression = '3*phi^2 - 2*phi^3'
    outputs = exodus
  [../]
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'
  end_time = 10
[]

[Outputs]
  exodus = true
[]

(modules/optimization/test/tests/optimizationreporter/nonlinear_material/forward_and_adjoint.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmax = 2
    ymax = 2
  []
[]

[Problem]
  nl_sys_names = 'nl0 adjoint'
  kernel_coverage_check = false
[]

[Variables]
  [forwardT]
  []
  [adjointT]
    solver_sys = adjoint
  []
[]

[Kernels]
  [heat_conduction]
    type = ADMatDiffusion
    variable = forwardT
    diffusivity = 'conductivity'
  []
  [heat_source]
    type = ADBodyForce
    function = volumetric_heat_func
    variable = forwardT
  []
[]

[DiracKernels]
  [pt]
    type = ReporterPointSource
    variable = adjointT
    x_coord_name = measurement_locations/measurement_xcoord
    y_coord_name = measurement_locations/measurement_ycoord
    z_coord_name = measurement_locations/measurement_zcoord
    value_name   = measurement_locations/misfit_values
  []
[]

[Materials]
  [NonlinearConductivity]
    type = ADParsedMaterial
    property_name = conductivity
    expression = '10+500*forwardT'
    coupled_variables = 'forwardT'
  []
[]

[Functions]
  [volumetric_heat_func]
    type = ParsedOptimizationFunction
    expression = q
    param_symbol_names = 'q'
    param_vector_name = 'params/heat_source'
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = forwardT
    boundary = left
    value = 0
  []
  [right]
    type = NeumannBC
    variable = forwardT
    boundary = right
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = forwardT
    boundary = bottom
    value = 2
  []
  [top]
    type = DirichletBC
    variable = forwardT
    boundary = top
    value = 1
  []
[]

[Reporters]
  [measurement_locations]
    type = OptimizationData
    variable = forwardT
  []
  [params]
    type = ConstantReporter
    real_vector_names = 'heat_source'
    real_vector_values = '0' # Dummy
  []
[]

[VectorPostprocessors]
  [gradient_vpp]
    type = ElementOptimizationSourceFunctionInnerProduct
    function = volumetric_heat_func
    variable = adjointT
    execute_on = ADJOINT_TIMESTEP_END
  []
[]

[Executioner]
  type = SteadyAndAdjoint
  forward_system = nl0
  adjoint_system = adjoint
  line_search = none
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-12
  l_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  console = false
[]

(modules/combined/test/tests/surface_tension_KKS/surface_tension_VDWgas.i)

# Test for ComputeExtraStressVDWGas
# Gas bubble with r = 15 nm in a solid matrix
# The gas pressure is counterbalanced by the surface tension of the solid-gas interface,
# which is included with ComputeSurfaceTensionKKS

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 300
  xmin = 0
  xmax = 30
[]

[Problem]
  coord_type = RSPHERICAL
[]

[GlobalParams]
  displacements = 'disp_x'
[]

[Variables]
  # order parameter
  [./eta]
    order = FIRST
    family = LAGRANGE
  [../]

  # gas concentration
  [./cg]
    order = FIRST
    family = LAGRANGE
  [../]

  # vacancy concentration
  [./cv]
    order = FIRST
    family = LAGRANGE
  [../]


  # gas chemical potential
  [./wg]
    order = FIRST
    family = LAGRANGE
  [../]

  # vacancy chemical potential
  [./wv]
    order = FIRST
    family = LAGRANGE
  [../]


  # Matrix phase gas concentration
  [./cgm]
    order = FIRST
    family = LAGRANGE
    initial_condition = 1.01e-31
  [../]

  # Matrix phase vacancy concentration
  [./cvm]
    order = FIRST
    family = LAGRANGE
    initial_condition = 2.25e-11
  [../]

  # Bubble phase gas concentration
  [./cgb]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.2714
  [../]

  # Bubble phase vacancy concentration
  [./cvb]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.7286
  [../]
[]

[ICs]
  [./eta_ic]
    variable = eta
    type = FunctionIC
    function = ic_func_eta
  [../]
  [./cv_ic]
    variable = cv
    type = FunctionIC
    function = ic_func_cv
  [../]
  [./cg_ic]
    variable = cg
    type = FunctionIC
    function = ic_func_cg
  [../]
[]

[Functions]
  [./ic_func_eta]
    type = ParsedFunction
    expression = 'r:=sqrt(x^2+y^2+z^2);0.5*(1.0-tanh((r-r0)/delta_eta/sqrt(2.0)))'
    symbol_names = 'delta_eta r0'
    symbol_values = '0.321     15'
  [../]
  [./ic_func_cv]
    type = ParsedFunction
    expression = 'r:=sqrt(x^2+y^2+z^2);eta_an:=0.5*(1.0-tanh((r-r0)/delta/sqrt(2.0)));cvbubinit*eta_an^3*(6*eta_an^2-15*eta_an+10)+cvmatrixinit*(1-eta_an^3*(6*eta_an^2-15*eta_an+10))'
    symbol_names = 'delta r0  cvbubinit cvmatrixinit'
    symbol_values = '0.321 15  0.7286    2.25e-11'
  [../]
  [./ic_func_cg]
    type = ParsedFunction
    expression = 'r:=sqrt(x^2+y^2+z^2);eta_an:=0.5*(1.0-tanh((r-r0)/delta/sqrt(2.0)));cgbubinit*eta_an^3*(6*eta_an^2-15*eta_an+10)+cgmatrixinit*(1-eta_an^3*(6*eta_an^2-15*eta_an+10))'
    symbol_names = 'delta r0  cgbubinit cgmatrixinit'
    symbol_values = '0.321 15  0.2714    1.01e-31'
  [../]
[]



[Modules/TensorMechanics/Master]
  [./all]
    add_variables = true
    generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz'
  [../]
[]

[Kernels]
  # enforce cg = (1-h(eta))*cgm + h(eta)*cgb
  [./PhaseConc_g]
    type = KKSPhaseConcentration
    ca       = cgm
    variable = cgb
    c        = cg
    eta      = eta
  [../]

  # enforce cv = (1-h(eta))*cvm + h(eta)*cvb
  [./PhaseConc_v]
    type = KKSPhaseConcentration
    ca       = cvm
    variable = cvb
    c        = cv
    eta      = eta
  [../]

  # enforce pointwise equality of chemical potentials
  [./ChemPotVacancies]
    type = KKSPhaseChemicalPotential
    variable = cvm
    cb       = cvb
    fa_name  = f_total_matrix
    fb_name  = f_total_bub
    args_a = 'cgm'
    args_b = 'cgb'
  [../]
  [./ChemPotGas]
    type = KKSPhaseChemicalPotential
    variable = cgm
    cb       = cgb
    fa_name  = f_total_matrix
    fb_name  = f_total_bub
    args_a = 'cvm'
    args_b = 'cvb'
  [../]

  #
  # Cahn-Hilliard Equations
  #
  [./CHBulk_g]
    type = KKSSplitCHCRes
    variable = cg
    ca       = cgm
    fa_name  = f_total_matrix
    w        = wg
    args_a   = 'cvm'
  [../]
  [./CHBulk_v]
    type = KKSSplitCHCRes
    variable = cv
    ca       = cvm
    fa_name  = f_total_matrix
    w        = wv
    args_a   = 'cgm'
  [../]

  [./dcgdt]
    type = CoupledTimeDerivative
    variable = wg
    v = cg
  [../]
  [./dcvdt]
    type = CoupledTimeDerivative
    variable = wv
    v = cv
  [../]

  [./wgkernel]
    type = SplitCHWRes
    mob_name = M
    variable = wg
  [../]
  [./wvkernel]
    type = SplitCHWRes
    mob_name = M
    variable = wv
  [../]

  #
  # Allen-Cahn Equation
  #
  [./ACBulkF]
    type = KKSACBulkF
    variable = eta
    fa_name  = f_total_matrix
    fb_name  = f_total_bub
    w        = 0.356
    args = 'cvm cvb cgm cgb'
  [../]
  [./ACBulkCv]
    type = KKSACBulkC
    variable = eta
    ca       = cvm
    cb       = cvb
    fa_name  = f_total_matrix
    args     = 'cgm'
  [../]
  [./ACBulkCg]
    type = KKSACBulkC
    variable = eta
    ca       = cgm
    cb       = cgb
    fa_name  = f_total_matrix
    args     = 'cvm'
  [../]
  [./ACInterface]
    type = ACInterface
    variable = eta
    kappa_name = kappa
  [../]
  [./detadt]
    type = TimeDerivative
    variable = eta
  [../]
[]

[Materials]
  # Chemical free energy of the matrix
  [./fm]
    type = DerivativeParsedMaterial
    property_name = fm
    coupled_variables = 'cvm cgm'
    material_property_names = 'kvmatrix kgmatrix cvmatrixeq cgmatrixeq'
    expression = '0.5*kvmatrix*(cvm-cvmatrixeq)^2 + 0.5*kgmatrix*(cgm-cgmatrixeq)^2'
  [../]
# Elastic energy of the matrix
  [./elastic_free_energy_m]
    type = ElasticEnergyMaterial
    base_name = matrix
    f_name = fe_m
    args = ' '
  [../]
# Total free energy of the matrix
  [./Total_energy_matrix]
    type = DerivativeSumMaterial
    property_name = f_total_matrix
    sum_materials = 'fm fe_m'
    coupled_variables = 'cvm cgm'
  [../]

  # Free energy of the bubble phase
  [./fb]
    type = DerivativeParsedMaterial
    property_name = fb
    coupled_variables = 'cvb cgb'
    material_property_names = 'kToverV nQ Va b f0 kpen kgbub kvbub cvbubeq cgbubeq'
    expression = '0.5*kgbub*(cvb-cvbubeq)^2 + 0.5*kvbub*(cgb-cgbubeq)^2'
  [../]

# Elastic energy of the bubble
  [./elastic_free_energy_p]
    type = ElasticEnergyMaterial
    base_name = bub
    f_name = fe_b
    args = ' '
  [../]

# Total free energy of the bubble
  [./Total_energy_bub]
    type = DerivativeSumMaterial
    property_name = f_total_bub
    sum_materials = 'fb fe_b'
    # sum_materials = 'fb'
    coupled_variables = 'cvb cgb'
  [../]

  # h(eta)
  [./h_eta]
    type = SwitchingFunctionMaterial
    h_order = HIGH
    eta = eta
  [../]

  # g(eta)
  [./g_eta]
    type = BarrierFunctionMaterial
    g_order = SIMPLE
    eta = eta
  [../]

  # constant properties
  [./constants]
    type = GenericConstantMaterial
    prop_names  = 'M   L   kappa  Va      kvmatrix  kgmatrix  kgbub kvbub f0      kpen  cvbubeq cgbubeq b      T'
    prop_values = '0.7 0.7 0.0368 0.03629 223.16    223.16    2.23  2.23  0.0224  1.0   0.6076  0.3924  0.085  800'
  [../]
  [./cvmatrixeq]
    type = ParsedMaterial
    property_name = cvmatrixeq
    material_property_names = 'T'
    constant_names        = 'kB           Efv'
    constant_expressions  = '8.6173324e-5 1.69'
    expression = 'exp(-Efv/(kB*T))'
  [../]
  [./cgmatrixeq]
    type = ParsedMaterial
    property_name = cgmatrixeq
    material_property_names = 'T'
    constant_names        = 'kB           Efg'
    constant_expressions  = '8.6173324e-5 4.92'
    expression = 'exp(-Efg/(kB*T))'
  [../]
  [./kToverV]
    type = ParsedMaterial
    property_name = kToverV
    material_property_names = 'T Va'
    constant_names        = 'k          C44dim' #k in J/K and dimensional C44 in J/m^3
    constant_expressions  = '1.38e-23   63e9'
    expression = 'k*T*1e27/Va/C44dim'
  [../]
  [./nQ]
    type = ParsedMaterial
    property_name = nQ
    material_property_names = 'T'
    constant_names        = 'k          Pi      M         hbar' #k in J/K, M is Xe atomic mass in kg, hbar in J s
    constant_expressions  = '1.38e-23   3.14159 2.18e-25  1.05459e-34'
    expression = '(M*k*T/2/Pi/hbar^2)^1.5 * 1e-27' #1e-27 converts from #/m^3 to #/nm^3
  [../]

  #Mechanical properties
  [./Stiffness_matrix]
    type = ComputeElasticityTensor
    C_ijkl = '0.778 0.7935'
    fill_method = symmetric_isotropic
    base_name = matrix
  [../]
  [./Stiffness_bub]
    type = ComputeElasticityTensor
    C_ijkl = '0.0778 0.07935'
    fill_method = symmetric_isotropic
    base_name = bub
  [../]
  [./strain_matrix]
    type = ComputeRSphericalSmallStrain
    base_name = matrix
  [../]
  [./strain_bub]
    type = ComputeRSphericalSmallStrain
    base_name = bub
  [../]
  [./stress_matrix]
    type = ComputeLinearElasticStress
    base_name = matrix
  [../]
  [./stress_bub]
    type = ComputeLinearElasticStress
    base_name = bub
  [../]
  [./global_stress]
    type = TwoPhaseStressMaterial
    base_A = matrix
    base_B = bub
  [../]
  [./surface_tension]
    type = ComputeSurfaceTensionKKS
    v = eta
    kappa_name = kappa
    w = 0.356
  [../]
  [./gas_pressure]
    type = ComputeExtraStressVDWGas
    T = T
    b = b
    cg = cgb
    Va = Va
    nondim_factor = 63e9
    base_name = bub
    outputs = exodus
  [../]
[]

[BCs]
  [./left_r]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  [../]
[]

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

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type   -sub_pc_factor_shift_type'
  petsc_options_value = 'asm       lu            nonzero'

  l_max_its = 30
  nl_max_its = 15
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1e-11
  num_steps = 2
  dt = 0.5
[]

[Outputs]
  exodus = true
[]

(modules/combined/examples/publications/rapid_dev/fig7a.i)

#
# Fig. 7 input for 10.1016/j.commatsci.2017.02.017
# D. Schwen et al./Computational Materials Science 132 (2017) 36-45
# Solid gray curve (1)
# Eigenstrain and elastic energies ar computed per phase and then interpolated.
# Supply the RADIUS parameter (10-35) on the command line to generate data
# for all curves in the plot.
#

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 32
  xmin = 0
  xmax = 100
  second_order = true
[]

[Problem]
  coord_type = RSPHERICAL
[]

[GlobalParams]
  displacements = 'disp_r'
[]

[Functions]
  [./diff]
    type = ParsedFunction
    expression = '${RADIUS}-pos_c'
    symbol_names = pos_c
    symbol_values = pos_c
  [../]
[]

# AuxVars to compute the free energy density for outputting
[AuxVariables]
  [./local_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./cross_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[AuxKernels]
  [./local_free_energy]
    type = TotalFreeEnergy
    variable = local_energy
    interfacial_vars = 'c'
    kappa_names = 'kappa_c'
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

[Variables]
  # Solute concentration variable
  [./c]
    [./InitialCondition]
      type = SmoothCircleIC
      invalue = 1
      outvalue = 0
      x1 = 0
      y1 = 0
      radius = ${RADIUS}
      int_width = 3
    [../]
  [../]
  [./w]
  [../]

  # Phase order parameter
  [./eta]
    [./InitialCondition]
      type = SmoothCircleIC
      invalue = 1
      outvalue = 0
      x1 = 0
      y1 = 0
      radius = ${RADIUS}
      int_width = 3
    [../]
  [../]

  # Mesh displacement
  [./disp_r]
    order = SECOND
  [../]

  [./Fe_fit]
    order = SECOND
  [../]
[]

[Kernels]
  # Set up stress divergence kernels
  [./TensorMechanics]
  [../]

  # Split Cahn-Hilliard kernels
  [./c_res]
    type = SplitCHParsed
    variable = c
    f_name = F
    args = 'eta'
    kappa_name = kappa_c
    w = w
  [../]
  [./wres]
    type = SplitCHWRes
    variable = w
    mob_name = M
  [../]
  [./time]
    type = CoupledTimeDerivative
    variable = w
    v = c
  [../]

  # Allen-Cahn and Lagrange-multiplier constraint kernels for order parameter 1
  [./detadt]
    type = TimeDerivative
    variable = eta
  [../]
  [./ACBulk1]
    type = AllenCahn
    variable = eta
    args = 'c'
    mob_name = L
    f_name = F
  [../]
  [./ACInterface]
    type = ACInterface
    variable = eta
    mob_name = L
    kappa_name = kappa_eta
  [../]

  [./Fe]
    type = MaterialPropertyValue
    prop_name = Fe
    variable = Fe_fit
  [../]

  [./autoadjust]
    type = MaskedBodyForce
    variable = w
    function = diff
    mask = mask
  [../]
[]

[Materials]
  # declare a few constants, such as mobilities (L,M) and interface gradient prefactors (kappa*)
  [./consts]
    type = GenericConstantMaterial
    prop_names  = 'M   L   kappa_c kappa_eta'
    prop_values = '1.0 1.0 0.5     1'
  [../]

  # forcing function mask
  [./mask]
    type = ParsedMaterial
    property_name = mask
    expression = grad/dt
    material_property_names = 'grad dt'
  [../]
  [./grad]
    type = VariableGradientMaterial
    variable = c
    prop = grad
  [../]
  [./time]
    type = TimeStepMaterial
  [../]

  # global mechanical properties
  [./elasticity_tensor_1]
    type = ComputeElasticityTensor
    C_ijkl = '1 1'
    base_name = phase1
    fill_method = symmetric_isotropic
  [../]
  [./elasticity_tensor_2]
    type = ComputeElasticityTensor
    C_ijkl = '1 1'
    base_name = phase2
    fill_method = symmetric_isotropic
  [../]
  [./strain_1]
    type = ComputeRSphericalSmallStrain
    base_name = phase1
  [../]
  [./strain_2]
    type = ComputeRSphericalSmallStrain
    base_name = phase2
    eigenstrain_names = eigenstrain
  [../]
  [./stress_1]
    type = ComputeLinearElasticStress
    base_name = phase1
  [../]
  [./stress_2]
    type = ComputeLinearElasticStress
    base_name = phase2
  [../]

  # eigenstrain per phase
  [./eigenstrain2]
    type = ComputeEigenstrain
    eigen_base = '0.05 0.05 0.05 0 0 0'
    base_name = phase2
    eigenstrain_name = eigenstrain
  [../]

  # switching functions
  [./switching]
    type = SwitchingFunctionMaterial
    function_name = h
    eta = eta
    h_order = SIMPLE
  [../]
  [./barrier]
    type = BarrierFunctionMaterial
    eta = eta
  [../]

  # chemical free energies
  [./chemical_free_energy_1]
    type = DerivativeParsedMaterial
    property_name = Fc1
    expression = 'c^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./chemical_free_energy_2]
    type = DerivativeParsedMaterial
    property_name = Fc2
    expression = '(1-c)^2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # elastic free energies
  [./elastic_free_energy_1]
    type = ElasticEnergyMaterial
    f_name = Fe1
    args = ''
    base_name = phase1
    derivative_order = 2
  [../]
  [./elastic_free_energy_2]
    type = ElasticEnergyMaterial
    f_name = Fe2
    args = ''
    base_name = phase2
    derivative_order = 2
  [../]

  # per phase free energies
  [./free_energy_1]
    type = DerivativeSumMaterial
    property_name = F1
    sum_materials = 'Fc1 Fe1'
    coupled_variables = 'c'
    derivative_order = 2
  [../]
  [./free_energy_2]
    type = DerivativeSumMaterial
    property_name = F2
    sum_materials = 'Fc2 Fe2'
    coupled_variables = 'c'
    derivative_order = 2
  [../]

  # global chemical free energy
  [./global_free_energy]
    type = DerivativeTwoPhaseMaterial
    f_name = F
    fa_name = F1
    fb_name = F2
    eta = eta
    args = 'c'
    W = 4
  [../]

  # global stress
  [./global_stress]
    type = TwoPhaseStressMaterial
    base_A = phase1
    base_B = phase2
  [../]

  [./elastic_free_energy]
    type = DerivativeTwoPhaseMaterial
    f_name = Fe
    fa_name = Fe1
    fb_name = Fe2
    eta = eta
    args = 'c'
    W = 0
  [../]
[]

[BCs]
  [./left_r]
    type = DirichletBC
    variable = disp_r
    boundary = 'left'
    value = 0
  [../]
[]

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

# We monitor the total free energy and the total solute concentration (should be constant)
[Postprocessors]
  [./total_free_energy]
    type = ElementIntegralVariablePostprocessor
    variable = local_energy
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./total_solute]
    type = ElementIntegralVariablePostprocessor
    variable = c
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./pos_c]
    type = FindValueOnLine
    start_point = '0 0 0'
    end_point = '100 0 0'
    v = c
    target = 0.582
    tol = 1e-8
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./pos_eta]
    type = FindValueOnLine
    start_point = '0 0 0'
    end_point = '100 0 0'
    v = eta
    target = 0.5
    tol = 1e-8
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
  [./c_min]
    type = ElementExtremeValue
    value_type = min
    variable = c
    execute_on = 'INITIAL TIMESTEP_END'
    outputs = 'table console'
  [../]
[]

[VectorPostprocessors]
  [./line]
    type = LineValueSampler
    variable = 'Fe_fit c w'
    start_point = '0 0 0'
    end_point =   '100 0 0'
    num_points = 5000
    sort_by = x
    outputs = vpp
    execute_on = 'INITIAL TIMESTEP_END'
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu'

  l_max_its = 30
  nl_max_its = 15
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-8
  nl_abs_tol = 2.0e-9
  start_time = 0.0
  end_time = 100000.0

  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 7
    iteration_window = 1
    dt = 1
  [../]

  [./Adaptivity]
    initial_adaptivity = 5
    interval = 10
    max_h_level = 5
    refine_fraction = 0.9
    coarsen_fraction = 0.1
  [../]
[]

[Outputs]
  print_linear_residuals = false
  perf_graph = true
  execute_on = 'INITIAL TIMESTEP_END'
  [./table]
    type = CSV
    delimiter = ' '
    file_base = radius_${RADIUS}/energy_pp
  [../]
  [./vpp]
    type = CSV
    delimiter = ' '
    sync_times = '10 50 100 500 1000 5000 10000 50000 100000'
    sync_only = true
    time_data = true
    file_base = radius_${RADIUS}/energy_vpp
  [../]
[]

(modules/solid_mechanics/test/tests/crystal_plasticity/cp_eigenstrains/hcp_volumetric_eigenstrain_decrease.i)

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 2
  ny = 2
  nz = 2
  elem_type = HEX8
[]

[AuxVariables]
  [temperature]
    order = FIRST
    family = LAGRANGE
  []
  [e_void_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [e_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [f_void_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [fp_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [resolved_shear_stress_3]
    order = CONSTANT
    family = MONOMIAL
  []
  [resolved_shear_stress_4]
    order = CONSTANT
    family = MONOMIAL
  []
  [resolved_shear_stress_9]
    order = CONSTANT
    family = MONOMIAL
  []
  [resolved_shear_stress_14]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = FINITE
  incremental = true
  add_variables = true
[]

[AuxKernels]
  [temperature]
    type = FunctionAux
    variable = temperature
    function = '300+400*t' # temperature increases at a constant rate
    execute_on = timestep_begin
  []
  [e_void_xx]
    type = RankTwoAux
    variable = e_void_xx
    rank_two_tensor = void_eigenstrain
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [e_void_yy]
    type = RankTwoAux
    variable = e_void_yy
    rank_two_tensor = void_eigenstrain
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [e_void_zz]
    type = RankTwoAux
    variable = e_void_zz
    rank_two_tensor = void_eigenstrain
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [f_void_zz]
    type = RankTwoAux
    variable = f_void_zz
    rank_two_tensor = volumetric_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [fp_zz]
    type = RankTwoAux
    variable = fp_zz
    rank_two_tensor = plastic_deformation_gradient
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [tau_3]
    type = MaterialStdVectorAux
    variable = resolved_shear_stress_3
    property = applied_shear_stress
    index = 3
    execute_on = timestep_end
  []
  [tau_4]
    type = MaterialStdVectorAux
    variable = resolved_shear_stress_4
    property = applied_shear_stress
    index = 4
    execute_on = timestep_end
  []
  [tau_9]
    type = MaterialStdVectorAux
    variable = resolved_shear_stress_9
    property = applied_shear_stress
    index = 9
    execute_on = timestep_end
  []
  [tau_14]
    type = MaterialStdVectorAux
    variable = resolved_shear_stress_14
    property = applied_shear_stress
    index = 14
    execute_on = timestep_end
  []
[]

[BCs]
  [symmy]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
  [symmx]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [symmz]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0
  []
  [tdisp]
    type = DirichletBC
    variable = disp_z
    boundary = front
    value = 0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeElasticityTensorCP
    C_ijkl = '1.684e5 1.214e5 1.214e5 1.684e5 1.214e5 1.684e5 0.754e5 0.754e5 0.754e5'
    fill_method = symmetric9
  []
  [stress]
    type = ComputeMultipleCrystalPlasticityStress
    crystal_plasticity_models = 'trial_xtalpl'
    eigenstrain_names = void_eigenstrain
    tan_mod_type = exact
    maximum_substep_iteration = 5
  []
  [trial_xtalpl]
    type = CrystalPlasticityHCPDislocationSlipBeyerleinUpdate
    number_slip_systems = 15
    slip_sys_file_name = hcp_aprismatic_capyramidal_slip_sys.txt
    unit_cell_dimension = '2.934e-7 2.934e-7 4.657e-7' #Ti, in mm, https://materialsproject.org/materials/mp-46/
    temperature = temperature
    initial_forest_dislocation_density = 15.0e3
    initial_substructure_density = 1.0e3
    slip_system_modes = 2
    number_slip_systems_per_mode = '3 12'
    lattice_friction_per_mode = '9 22' #Knezevic et al MSEA 654 (2013)
    effective_shear_modulus_per_mode = '4.7e2 4.7e2' #Ti, in MPa, https://materialsproject.org/materials/mp-46/
    burgers_vector_per_mode = '2.934e-7 6.586e-7' #Ti, in mm, https://materialsproject.org/materials/mp-46/
    slip_generation_coefficient_per_mode = '1.25e5 2.25e7' #from Beyerlein and Tome 2008 IJP
    normalized_slip_activiation_energy_per_mode = '3.73e-3 3.2e-2' #from Beyerlein and Tome 2008 IJP
    slip_energy_proportionality_factor_per_mode = '330 100' #from Beyerlein and Tome 2008 IJP
    substructure_rate_coefficient_per_mode = '355 0.4' #from Capolungo et al MSEA (2009)
    applied_strain_rate = 0.001
    gamma_o = 1.0e-3
    Hall_Petch_like_constant_per_mode = '0.2 0.2' #Estimated to match graph in Capolungo et al MSEA (2009), Figure 2
    grain_size = 20.0e-3 #20 microns, Beyerlein and Tome IJP (2008)
  []
  [void_eigenstrain]
    type = ComputeCrystalPlasticityVolumetricEigenstrain
    eigenstrain_name = void_eigenstrain
    deformation_gradient_name = volumetric_deformation_gradient
    mean_spherical_void_radius = void_radius
    spherical_void_number_density = void_density
  []
  [void_radius]
    type = ParsedMaterial
    property_name = void_radius
    coupled_variables = temperature
    expression = 'if(temperature<321.0, 1.0e-5, (1.0e-5  - 5.0e-8 * (temperature - 320)))' #mm
  []
  [void_density]
    type = GenericConstantMaterial
    prop_names = void_density
    prop_values = '1.0e8'  ###1/mm^3
  []
[]

[Postprocessors]
  [e_void_xx]
    type = ElementAverageValue
    variable = e_void_xx
  []
  [e_void_yy]
    type = ElementAverageValue
    variable = e_void_yy
  []
  [e_void_zz]
    type = ElementAverageValue
    variable = e_void_zz
  []
  [f_void_zz]
    type = ElementAverageValue
    variable = f_void_zz
  []
  [void_density]
    type = ElementAverageMaterialProperty
    mat_prop = void_density
    execute_on = TIMESTEP_END
  []
  [void_radius]
    type = ElementAverageMaterialProperty
    mat_prop = void_radius
    execute_on = TIMESTEP_END
  []
  [fp_zz]
    type = ElementAverageValue
    variable = fp_zz
  []
  [tau_3]
    type = ElementAverageValue
    variable = resolved_shear_stress_3
  []
  [tau_4]
    type = ElementAverageValue
    variable = resolved_shear_stress_4
  []
  [tau_9]
    type = ElementAverageValue
    variable = resolved_shear_stress_9
  []
  [tau_14]
    type = ElementAverageValue
    variable = resolved_shear_stress_14
  []
[]

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

[Debug]
  show_var_residual_norms = true
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'

  petsc_options = '-snes_converged_reason'
  petsc_options_iname = '-pc_type -pc_asm_overlap -sub_pc_type -ksp_type -ksp_gmres_restart'
  petsc_options_value = ' asm      2              lu            gmres     200'
  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-8
  nl_abs_step_tol = 1e-10

  dt = 0.05
  dtmin = 1e-4
  num_steps = 10
[]

[Outputs]
  csv = true
  perf_graph = true
[]

(modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialPFM.i)

# this input file test the implementation of the grand-potential phase-field model based on M.Plapp PRE 84,031601(2011)
# in this simple example, the liquid and solid free energies are parabola with the same curvature and the material properties are constant
# Note that this example also test The SusceptibilityTimeDerivative kernels
[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 16
  ny = 16
  xmax = 32
  ymax = 32
[]

[GlobalParams]
  radius = 20.0
  int_width = 4.0
  x1 = 0
  y1 = 0
[]

[Variables]
  [./w]
  [../]
  [./eta]
  [../]
[]

[ICs]
  [./w]
    type = SmoothCircleIC
    variable = w
    # note w = A*(c-cleq), A = 1.0, cleq = 0.0 ,i.e., w = c (in the matrix/liquid phase)
    outvalue = -0.2
    invalue = 0.2
  [../]
  [./eta]
    type = SmoothCircleIC
    variable = eta
    outvalue = 0.0
    invalue = 1.0
  [../]
[]

[Kernels]
  [./w_dot]
    type = SusceptibilityTimeDerivative
    variable = w
    f_name = chi
    coupled_variables = '' # in this case chi (the susceptibility) is simply a constant
  [../]
  [./Diffusion]
    type = MatDiffusion
    variable = w
    diffusivity = D
    args = ''
  [../]
  [./coupled_etadot]
    type = CoupledSusceptibilityTimeDerivative
    variable = w
    v = eta
    f_name = ft
    coupled_variables = 'eta'
  [../]
  [./AC_bulk]
    type = AllenCahn
    variable = eta
    f_name = F
    coupled_variables = 'w'
  [../]
  [./AC_int]
    type = ACInterface
    variable = eta
  [../]
  [./e_dot]
    type = TimeDerivative
    variable = eta
  [../]
[]

[Materials]
  [./constants]
    type = GenericConstantMaterial
    prop_names  = 'kappa_op  D    L    chi  cseq   cleq   A'
    prop_values = '4.0       1.0  1.0  1.0  0.0  1.0  1.0'
  [../]

  [./liquid_GrandPotential]
    type = DerivativeParsedMaterial
    expression = '-0.5 * w^2/A - cleq * w'
    coupled_variables = 'w'
    property_name = f1
    material_property_names = 'cleq A'
  [../]
  [./solid_GrandPotential]
    type = DerivativeParsedMaterial
    expression = '-0.5 * w^2/A - cseq * w'
    coupled_variables = 'w'
    property_name = f2
    material_property_names = 'cseq A'
  [../]
  [./switching_function]
    type = SwitchingFunctionMaterial
    eta = eta
    h_order = HIGH
  [../]
  [./barrier_function]
    type = BarrierFunctionMaterial
    eta = eta
  [../]
  [./cs]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = cs
    material_property_names = 'A cseq'
    expression = 'w/A + cseq' # since w = A*(c-cseq)
    derivative_order = 2
  [../]
  [./cl]
    type = DerivativeParsedMaterial
    coupled_variables = 'w'
    property_name = cl
    material_property_names = 'A cleq'
    expression = 'w/A + cleq' # since w = A*(c-cleq)
    derivative_order = 2
  [../]
  [./total_GrandPotential]
    type = DerivativeTwoPhaseMaterial
    coupled_variables = 'w'
    eta = eta
    fa_name = f1
    fb_name = f2
    derivative_order = 2
    W = 1.0
  [../]
  [./coupled_eta_function]
    type = DerivativeParsedMaterial
    expression = '(cs - cl) * dh'
    coupled_variables = 'eta w'
    property_name = ft
    material_property_names = 'cs cl dh:=D[h,eta]'
    derivative_order = 1
    outputs = exodus
  [../]

  [./concentration]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'dF:=D[F,w]'
    expression = '-dF'
    outputs = exodus
  [../]
[]

[Postprocessors]
  [./C]
    type = ElementIntegralMaterialProperty
    mat_prop = c
    execute_on = 'initial timestep_end'
  [../]
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = NEWTON

  l_max_its = 15
  l_tol = 1e-3
  nl_max_its = 15
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8

  num_steps = 5
  dt = 10.0
[]

[Outputs]
  exodus = true
  csv = true
  execute_on = 'TIMESTEP_END'
[]

(test/tests/materials/ad_material/conversion/1d_dirichlet.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmax = 2
[]

[Variables]
  [v]
    initial_condition = 1.1
  []
[]

[Kernels]
  inactive = 'ad_diff'
  [diff]
    type = MatDiffusion
    variable = v
    diffusivity = 'coef'
  []
  [ad_diff]
    type = ADMatDiffusion
    variable = v
    diffusivity = 'ad_coef_2'
  []
  [sink]
    type = ADBodyForce
    variable = v
    function = 'sink'
  []
[]

[BCs]
  [bounds]
    type = DirichletBC
    variable = v
    boundary = 'left right'
    value = 0
  []
[]

[Functions]
  [sink]
    type = ParsedFunction
    expression = '3*x^3'
  []
[]

[Materials]
  [ad_coef]
    type = ADParsedMaterial
    property_name = 'ad_coef'
    expression = '0.01 * max(v, 1)'
    coupled_variables = 'v'
  []
  [converter_to_regular]
    type = MaterialADConverter
    ad_props_in = 'ad_coef'
    reg_props_out = 'coef'
  []
  # at this point we should have lost the derivatives
  [converter_to_ad]
    type = MaterialADConverter
    reg_props_in = 'coef'
    ad_props_out = 'ad_coef_2'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
[]

(test/tests/materials/derivative_material_interface/ad_construction_order.i)

#
# Test the the getDefaultMaterialProperty in DerivativeMaterialInterface.
# This test should only pass, if the construction order of the Materials
# using this interface does not influence the outcome.
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 5
  ny = 1
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 0.1
  elem_type = QUAD4
[]

[GlobalParams]
  derivative_order = 2
[]

[Variables]
  [./c]
    [./InitialCondition]
      type = FunctionIC
      function = x
    [../]
  [../]
[]

[Kernels]
  [./dummy1]
    type = ADDiffusion
    variable = c
  [../]
  [./dummy2]
    type = ADTimeDerivative
    variable = c
  [../]
[]

[Materials]
  # derivatives used both before and after being declared
  [./sum_a_1]
    type = ADDerivativeSumMaterial
    property_name = Fa1
    sum_materials = 'Fa'
    coupled_variables = 'c'
    outputs = exodus
  [../]
  [./free_energy_a]
    type = ADDerivativeParsedMaterial
    property_name = Fa
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_a_2]
    type = ADDerivativeSumMaterial
    property_name = Fa2
    sum_materials = 'Fa'
    coupled_variables = 'c'
    outputs = exodus
  [../]

  # derivatives declared after being used
  [./sum_b_1]
    type = ADDerivativeSumMaterial
    property_name = Fb1
    sum_materials = 'Fb'
    coupled_variables = 'c'
    outputs = exodus
  [../]
  [./free_energy_b]
    type = ADDerivativeParsedMaterial
    property_name = Fb
    coupled_variables = 'c'
    expression = 'c^4'
  [../]

  # derivatives declared before being used
  [./free_energy_c]
    type = ADDerivativeParsedMaterial
    property_name = Fc
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_c_2]
    type = ADDerivativeSumMaterial
    property_name = Fc2
    sum_materials = 'Fc'
    coupled_variables = 'c'
    outputs = exodus
  [../]

  # non-existing derivatives
  [./free_energy_d]
    type = ADParsedMaterial
    property_name = Fd
    coupled_variables = 'c'
    expression = 'c^4'
  [../]
  [./sum_d_1]
    type = ADDerivativeSumMaterial
    property_name = Fd1
    sum_materials = 'Fd'
    coupled_variables = 'c'
    outputs = exodus
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2

  solve_type = 'NEWTON'
  num_steps = 1
  dt = 1e-5
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
[]

(modules/optimization/test/tests/executioners/constrained/shape_optimization/forward.i)

# This test is documented as an example for ConstrainedShapeOptimization. This
# test should not be changed without updating the documentation.

inner_radius = 6
outer_radius = 10
volume_constraint = 200

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = ConcentricCircleMeshGenerator
    has_outer_square = no
    num_sectors = 16
    radii = '${inner_radius} ${outer_radius}'
    rings = '16 16'
    preserve_volumes = false
  []
  [inner_radius]
    type = SideSetsBetweenSubdomainsGenerator
    input = mesh
    new_boundary = inner
    primary_block = 2
    paired_block = 1
  []

  [delete]
    type = BlockDeletionGenerator
    input = inner_radius
    block = 1
  []
  [gather_all]
    type = BoundingBoxNodeSetGenerator
    input = delete
    bottom_left = '-100 -100 -100'
    top_right = '100 100 100'
    new_boundary = total
  []
  [combine]
    type = SideSetsFromBoundingBoxGenerator
    input = gather_all
    bottom_left = '-100 -100 -100'
    top_right = '100 100 100'
    boundaries_old = 'inner outer'
    boundary_new = moving
  []

[]
[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [T]
  []
[]

[AuxVariables]
  [dist_between]
    [AuxKernel]
      type = NearestNodeDistanceAux
      variable = dist_between
      paired_boundary = moving
      boundary = total
      block = 2
      use_displaced_mesh = false
      execute_on = "INITIAL"
    []
  []
[]

[Kernels]
  [disp_x]
    type = MatDiffusion
    variable = disp_x
    use_displaced_mesh = false
    diffusivity = diff_coef
  []
  [disp_y]
    type = MatDiffusion
    variable = disp_y
    use_displaced_mesh = false
    diffusivity = diff_coef
  []
  # run physics of interest on deformed mesh
  [Diffusion]
    type = FunctionDiffusion
    variable = T
    use_displaced_mesh = true
  []
  [Source]
    type = BodyForce
    variable = T
    value = 1
    use_displaced_mesh = true
  []
[]

[Materials]
  # perserve elements near the boundary
  [diff_coef]
    type = ParsedMaterial
    coupled_variables = 'dist_between'
    expression = '1/(dist_between+0.5)'
    property_name = 'diff_coef'
  []
  [h]
    type = ADGenericFunctionMaterial
    prop_names = h
    prop_values = h
  []
  # convection type boundary condition
  [convection_bc]
    type = ADParsedMaterial
    coupled_variables = "T"
    expression = "h*(100-T)"
    material_property_names = "h"
    property_name = convection
  []
[]

[Functions]
  [r1_x]
    type = ParsedOptimizationFunction
    expression = 'r1 * cos((atan(y/x)))'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [r1_y]
    type = ParsedOptimizationFunction
    expression = 'r1 * sin((atan(y/x)))'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [r0_x]
    type = ParsedOptimizationFunction
    expression = 'r0 * cos((atan(y/x)))'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [r0_y]
    type = ParsedOptimizationFunction
    expression = 'r0 * sin((atan(y/x)))'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [h]
    type = ParsedOptimizationFunction
    # r0+${inner_radius} is the true current inner radius
    expression = '10 /(pi * (r0+${inner_radius})^3)'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [eq_grad_r0]
    type = ParsedOptimizationFunction
    expression = '-2 * pi * (r0 + ${inner_radius})'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
  [eq_grad_r1]
    type = ParsedOptimizationFunction
    # r1+${outer_radius} is the true current outer radius
    expression = '2 * pi * (r1+${outer_radius})'
    param_symbol_names = 'r0 r1'
    param_vector_name = 'params/radii'
  []
[]

[BCs]
  [diffuse_r1_x]
    type = ADFunctionDirichletBC
    variable = disp_x
    boundary = 'outer'
    function = r1_x
    preset = false
  []
  [diffuse_r1_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = 'outer'
    function = r1_y
    preset = false
  []
  [diffuse_r0_x]
    type = ADFunctionDirichletBC
    variable = disp_x
    boundary = 'inner'
    function = r0_x
    preset = false
  []
  [diffuse_r0_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = 'inner'
    function = r0_y
    preset = false
  []
  # run physics on deformed mesh
  [convection]
    type = ADMatNeumannBC
    variable = T
    boundary = inner
    boundary_material = convection
    use_displaced_mesh = true
    value = 1
  []
[]

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

[Reporters]
  [params]
    type = ConstantReporter
    real_vector_names = 'radii'
    real_vector_values = '0 0'
    dof_id_type_vector_names = 'num_params'
    dof_id_type_vector_values = '2'
  []
[]

[Postprocessors]
  [current_volume]
    type = VolumePostprocessor
    use_displaced_mesh = true
  []
  # objective function
  [objective]
    type = NodalExtremeValue
    variable = T
  []
  [eq_constraint]
    type = ParsedPostprocessor
    pp_names = current_volume
    expression = 'current_volume - ${volume_constraint}'
  []
  [func_r0]
    type = FunctionValuePostprocessor
    function = eq_grad_r0
  []
  [func_r1]
    type = FunctionValuePostprocessor
    function = eq_grad_r1
  []
[]

[VectorPostprocessors]
  # convert "Real" postprocessors to vectors
  [vol_constraint]
    type = VectorOfPostprocessors
    postprocessors = 'eq_constraint'
    force_postaux = true
  []
  [eq_grad]
    type = VectorOfPostprocessors
    postprocessors = 'func_r0 func_r1'
    force_postaux = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu       NONZERO'
  line_search = none
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8
[]

[Outputs]
  console = false
[]

(modules/combined/examples/optimization/helmholtz_multimat_strip.i)

vol_frac = 0.35

power = 1.1

Emin = 1.0e-6
Ess = 0.475 # ss
Et = 1.0 # w

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  # final_generator = 'MoveRight'
  [Bottom]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 320
    ny = 30
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 15
  []
  [RenameBottom]
    type = RenameBoundaryGenerator
    input = Bottom
    old_boundary = 'top bottom right left'
    new_boundary = 'top_bottom bottom_bottom right_bottom left_bottom'
  []
  [Middle]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 320
    ny = 6
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 3
  []
  [MoveMiddle]
    type = TransformGenerator
    input = Middle
    transform = TRANSLATE
    vector_value = '0 15 0'
  []
  [RenameMiddle]
    type = RenameBoundaryGenerator
    input = MoveMiddle
    old_boundary = 'top bottom right left'
    new_boundary = 'top_middle bottom_middle right_middle left_middle'
  []
  [Top]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 320
    ny = 30
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 15
  []
  [MoveTop]
    type = TransformGenerator
    input = Top
    transform = TRANSLATE
    vector_value = '0 18 0'
  []
  [RenameTop]
    type = RenameBoundaryGenerator
    input = MoveTop
    old_boundary = 'top bottom right left'
    new_boundary = 'top_top bottom_top right_top left_top'
  []
  [bottom_gen]
    type = ParsedSubdomainMeshGenerator
    input = RenameBottom
    combinatorial_geometry = 'y <= 15'
    block_id = 1
  []
  [middle_gen]
    type = ParsedSubdomainMeshGenerator
    input = RenameMiddle
    combinatorial_geometry = 'y <= 18 & y > 15'
    block_id = 2
  []
  [top_gen]
    type = ParsedSubdomainMeshGenerator
    input = RenameTop
    combinatorial_geometry = 'y > 18'
    block_id = 3
  []
  [stitch]
    type = StitchedMeshGenerator
    inputs = 'bottom_gen middle_gen top_gen'
    stitch_boundaries_pairs = 'top_bottom bottom_middle; top_middle bottom_top'
  []
  [left_load]
    type = ExtraNodesetGenerator
    input = stitch
    new_boundary = left_load
    coord = '37.5 33 0'
  []
  [right_load]
    type = ExtraNodesetGenerator
    input = left_load
    new_boundary = right_load
    coord = '112.5 33 0'
  []
  [left_support]
    type = ExtraNodesetGenerator
    input = right_load
    new_boundary = left_support
    coord = '0 0 0'
  []
  [right_support]
    type = ExtraNodesetGenerator
    input = left_support
    new_boundary = right_support
    coord = '150 0 0'
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [Dc]
    initial_condition = -1.0
  []
[]

[AuxVariables]
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []

  [sensitivity]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
    [AuxKernel]
      type = MaterialRealAux
      variable = sensitivity
      property = sensitivity
      execute_on = LINEAR
    []
    block = '1 2 3'
  []
  [mat_den_nodal]
    family = L2_LAGRANGE
    order = FIRST
    initial_condition = ${vol_frac}
    [AuxKernel]
      type = SelfAux
      execute_on = TIMESTEP_END
      variable = mat_den_nodal
      v = mat_den
    []
  []
  [Dc_elem]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
    [AuxKernel]
      type = SelfAux
      variable = Dc_elem
      v = Dc
      execute_on = 'TIMESTEP_END'
    []
  []
[]

[Kernels]
  [diffusion]
    type = FunctionDiffusion
    variable = Dc
    function = 4.0
  []
  [potential]
    type = Reaction
    variable = Dc
  []
  [source]
    type = CoupledForce
    variable = Dc
    v = sensitivity
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = left_support
    value = 0.0
  []
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left_support
    value = 0.0
  []
  [no_y_right]
    type = DirichletBC
    variable = disp_y
    boundary = right_support
    value = 0.0
  []
  [boundary_penalty]
    type = ADRobinBC
    variable = Dc
    boundary = 'bottom_bottom right_bottom left_bottom top_top right_top left_top left_middle '
               'right_middle'
    coefficient = 10
  []
[]

[NodalKernels]
  [left_down]
    type = NodalGravity
    variable = disp_y
    boundary = left_load
    gravity_value = -1e-3
    mass = 1
  []
  [right_down]
    type = NodalGravity
    variable = disp_y
    boundary = right_load
    gravity_value = -1e-3
    mass = 1
  []
[]

[Materials]
  [sensitivity]
    type = ParsedMaterial
    property_name = 'sensitivity'
    block = '2'
    expression = '0'
  []
  [elasticity_tensor_one]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys_one
    poissons_ratio = poissons_ratio
    args = 'mat_den'
    block = '1'
  []
  [elasticity_tensor_three]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys_three
    poissons_ratio = poissons_ratio
    args = 'mat_den'
    block = '3'
  []
  [elasticity_tensor_two]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1.0
    poissons_ratio = 0.3
    block = '2'
  []
  # One: Tungsten
  [E_phys_one]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${Et}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys_one
    block = '1'
    outputs = 'exodus'
  []
  # Three: SS316
  [E_phys_three]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${Ess}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys_three
    block = '3'
    outputs = 'exodus'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc_one]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys_one
    block = '1'
  []
  [dc_three]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys_three
    block = '3'
  []
[]

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

[UserObjects]
  [update_one]
    type = DensityUpdate
    density_sensitivity = Dc_elem
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
    block = '1'
  []
  [update_three]
    type = DensityUpdate
    density_sensitivity = Dc_elem
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
    block = '3'
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-10
  dt = 1.0
  num_steps = 90
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
    block = '1 3'
  []
  [objective_one]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
    block = '1'
  []
  [objective_three]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
    block = '3'
  []
[]

(modules/navier_stokes/test/tests/finite_element/ins/cg-dg-hybrid/lid-driven/hybrid-cg-dg.i)

mu = 1
rho = 1
l = 200
U = 1

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${l}
    ymin = 0
    ymax = ${l}
    nx = 20
    ny = 20
  []
  [corner_node]
    type = ExtraNodesetGenerator
    new_boundary = 'pinned_node'
    nodes = '0'
    input = gen
  []
[]

[Variables]
  [u]
    family = MONOMIAL
  []
  [v]
    family = MONOMIAL
  []
  [pressure][]
[]

[Kernels]
  [momentum_x_convection]
    type = ADConservativeAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = MatDiffusion
    variable = u
    diffusivity = 'mu'
  []
  [momentum_x_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = u
    pressure = pressure
    component = 0
  []
  [momentum_y_convection]
    type = ADConservativeAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = MatDiffusion
    variable = v
    diffusivity = 'mu'
  []
  [momentum_y_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = v
    pressure = pressure
    component = 1
  []
  [mass]
    type = ADConservativeAdvection
    variable = pressure
    velocity = velocity
    advected_quantity = -1
  []
[]

[DGKernels]
  [momentum_x_convection]
    type = ADDGAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = DGDiffusion
    variable = u
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
  [momentum_y_convection]
    type = ADDGAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = DGDiffusion
    variable = v
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
[]

[BCs]
  [u_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right'
    variable = u
    sigma = 6
    epsilon = -1
    function = '0'
    diff = 'mu'
  []
  [v_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = v
    sigma = 6
    epsilon = -1
    function = '0'
    diff = 'mu'
  []
  [u_top]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'top'
    variable = u
    sigma = 6
    epsilon = -1
    function = '${U}'
    diff = 'mu'
  []
  [pressure_pin]
    type = DirichletBC
    variable = pressure
    boundary = 'pinned_node'
    value = 0
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho'
    prop_values = '${rho}'
  []
  [const_reg]
    type = GenericConstantMaterial
    prop_names = 'mu'
    prop_values = '${mu}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = u
    v = v
  []
  [rhou]
    type = ADParsedMaterial
    property_name = 'rhou'
    coupled_variables = 'u'
    material_property_names = 'rho'
    expression = 'rho*u'
  []
  [rhov]
    type = ADParsedMaterial
    property_name = 'rhov'
    coupled_variables = 'v'
    material_property_names = 'rho'
    expression = 'rho*v'
  []
[]

[AuxVariables]
  [vel_x]
    family = MONOMIAL
    order = CONSTANT
  []
  [vel_y]
    family = MONOMIAL
    order = CONSTANT
  []
  [p]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [vel_x]
    type = ProjectionAux
    variable = vel_x
    v = u
  []
  [vel_y]
    type = ProjectionAux
    variable = vel_y
    v = v
  []
  [p]
    type = ProjectionAux
    variable = p
    v = pressure
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu       NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

[Postprocessors]
  [Re]
    type = ParsedPostprocessor
    expression = '${rho} * ${U} * ${l} / ${mu}'
  []
[]

(modules/optimization/test/tests/executioners/transient_and_adjoint/nonlinear_diffusivity.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 1
    ymax = 1
    nx = 10
    ny = 10
  []
[]

[Problem]
  nl_sys_names = 'nl0 adjoint'
[]

[Variables]
  [u]
  []
  [u_adjoint]
    solver_sys = adjoint
  []
[]

[Kernels]
  [time]
    type = TimeDerivative
    variable = u
  []
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = D
  []
  [src]
    type = ADBodyForce
    variable = u
    value = 1
  []

  [src_adjoint]
    type = ADBodyForce
    variable = u_adjoint
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = ADDirichletBC
    variable = u
    boundary = 'top right'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = ADParsedMaterial
    property_name = D
    expression = '0.1 + 5 * u'
    coupled_variables = 'u'
  []
[]

[Postprocessors]
  [u_avg]
    type = ElementAverageValue
    variable = u
    execute_on = 'TIMESTEP_END ADJOINT_TIMESTEP_END'
  []
  [u_adjoint_avg]
    type = ElementAverageValue
    variable = u_adjoint
    execute_on = ADJOINT_TIMESTEP_END
  []
  [inner_product]
    type = VariableInnerProduct
    variable = u
    second_variable = u_adjoint
    execute_on = ADJOINT_TIMESTEP_END
  []
[]

[Executioner]
  type = TransientAndAdjoint
  forward_system = nl0
  adjoint_system = adjoint

  dt = 0.2
  num_steps = 5

  nl_rel_tol = 1e-12
  l_tol = 1e-12
[]

[Outputs]
  [forward]
    type = CSV
  []
  [adjoint]
    type = CSV
    execute_on = 'INITIAL ADJOINT_TIMESTEP_END'
  []
  [console]
    type = Console
    execute_postprocessors_on = 'INITIAL TIMESTEP_END ADJOINT_TIMESTEP_END'
  []
[]

(modules/navier_stokes/test/tests/finite_element/ins/cg-dg-hybrid/mms/lid-driven/hybrid-cg-dg-mms.i)

rho=1.1
mu=1.1
cp=1.1
k=1.1

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = -1
    xmax = 1.0
    ymin = -1
    ymax = 1.0
    nx = 2
    ny = 2
  []
  [corner_node]
    type = ExtraNodesetGenerator
    new_boundary = 'pinned_node'
    nodes = '0'
    input = gen
  []
[]

[Variables]
  [u]
    family = MONOMIAL
  []
  [v]
    family = MONOMIAL
  []
  [pressure][]
  [T]
    family = MONOMIAL
  []
[]

[Kernels]
  [momentum_x_convection]
    type = ADConservativeAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = MatDiffusion
    variable = u
    diffusivity = 'mu'
  []
  [momentum_x_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = u
    pressure = pressure
    component = 0
  []
  [u_forcing]
    type = BodyForce
    variable = u
    function = forcing_u
  []
  [momentum_y_convection]
    type = ADConservativeAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = MatDiffusion
    variable = v
    diffusivity = 'mu'
  []
  [momentum_y_pressure]
    type = PressureGradient
    integrate_p_by_parts = false
    variable = v
    pressure = pressure
    component = 1
  []
  [v_forcing]
    type = BodyForce
    variable = v
    function = forcing_v
  []
  [mass]
    type = ADConservativeAdvection
    variable = pressure
    velocity = velocity
    advected_quantity = -1
  []
  [p_forcing]
    type = BodyForce
    variable = pressure
    function = forcing_p
  []
  [T_convection]
    type = ADConservativeAdvection
    variable = T
    velocity = 'velocity'
    advected_quantity = 'rho_cp_temp'
  []
  [T_diffusion]
    type = MatDiffusion
    variable = T
    diffusivity = 'k'
  []
  [T_forcing]
    type = BodyForce
    variable = T
    function = forcing_T
  []
[]

[DGKernels]
  [momentum_x_convection]
    type = ADDGAdvection
    variable = u
    velocity = 'velocity'
    advected_quantity = 'rhou'
  []
  [momentum_x_diffusion]
    type = DGDiffusion
    variable = u
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
  [momentum_y_convection]
    type = ADDGAdvection
    variable = v
    velocity = 'velocity'
    advected_quantity = 'rhov'
  []
  [momentum_y_diffusion]
    type = DGDiffusion
    variable = v
    sigma = 6
    epsilon = -1
    diff = 'mu'
  []
  [T_convection]
    type = ADDGAdvection
    variable = T
    velocity = 'velocity'
    advected_quantity = 'rho_cp_temp'
  []
  [T_diffusion]
    type = DGDiffusion
    variable = T
    sigma = 6
    epsilon = -1
    diff = 'k'
  []
[]

[BCs]
  [u_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = u
    sigma = 6
    epsilon = -1
    function = exact_u
    diff = 'mu'
  []
  [v_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = v
    sigma = 6
    epsilon = -1
    function = exact_v
    diff = 'mu'
  []
  [pressure_pin]
    type = FunctionDirichletBC
    variable = pressure
    boundary = 'pinned_node'
    function = 'exact_p'
  []
  [T_walls]
    type = DGFunctionDiffusionDirichletBC
    boundary = 'left bottom right top'
    variable = T
    sigma = 6
    epsilon = -1
    function = exact_T
    diff = 'k'
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho cp'
    prop_values = '${rho} ${cp}'
  []
  [const_reg]
    type = GenericConstantMaterial
    prop_names = 'mu k'
    prop_values = '${mu} ${k}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = u
    v = v
  []
  [rhou]
    type = ADParsedMaterial
    property_name = 'rhou'
    coupled_variables = 'u'
    material_property_names = 'rho'
    expression = 'rho*u'
  []
  [rhov]
    type = ADParsedMaterial
    property_name = 'rhov'
    coupled_variables = 'v'
    material_property_names = 'rho'
    expression = 'rho*v'
  []
  [rho_cp]
    type = ADParsedMaterial
    property_name = 'rho_cp'
    material_property_names = 'rho cp'
    expression = 'rho*cp'
  []
  [rho_cp_temp]
    type = ADParsedMaterial
    property_name = 'rho_cp_temp'
    material_property_names = 'rho cp'
    coupled_variables = 'T'
    expression = 'rho*cp*T'
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'sin(y)*cos((1/2)*x*pi)'
  []
  [forcing_u]
    type = ParsedFunction
    expression = 'mu*sin(y)*cos((1/2)*x*pi) + (1/4)*pi^2*mu*sin(y)*cos((1/2)*x*pi) - 1/2*pi*rho*sin(x)*sin(y)*sin((1/2)*y*pi)*cos((1/2)*x*pi) + rho*sin(x)*cos(y)*cos((1/2)*x*pi)*cos((1/2)*y*pi) - pi*rho*sin(y)^2*sin((1/2)*x*pi)*cos((1/2)*x*pi) + sin(y)*cos(x)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_v]
    type = ParsedFunction
    expression = 'sin(x)*cos((1/2)*y*pi)'
  []
  [forcing_v]
    type = ParsedFunction
    expression = 'mu*sin(x)*cos((1/2)*y*pi) + (1/4)*pi^2*mu*sin(x)*cos((1/2)*y*pi) - pi*rho*sin(x)^2*sin((1/2)*y*pi)*cos((1/2)*y*pi) - 1/2*pi*rho*sin(x)*sin(y)*sin((1/2)*x*pi)*cos((1/2)*y*pi) + rho*sin(y)*cos(x)*cos((1/2)*x*pi)*cos((1/2)*y*pi) + sin(x)*cos(y)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'sin(x)*sin(y)'
  []
  [forcing_p]
    type = ParsedFunction
    expression = '(1/2)*pi*sin(x)*sin((1/2)*y*pi) + (1/2)*pi*sin(y)*sin((1/2)*x*pi)'
  []
  [exact_T]
    type = ParsedFunction
    expression = 'cos(x)*cos(y)'
  []
  [forcing_T]
    type = ParsedFunction
    expression = '-cp*rho*sin(x)*sin(y)*cos(x)*cos((1/2)*y*pi) - cp*rho*sin(x)*sin(y)*cos(y)*cos((1/2)*x*pi) - 1/2*pi*cp*rho*sin(x)*sin((1/2)*y*pi)*cos(x)*cos(y) - 1/2*pi*cp*rho*sin(y)*sin((1/2)*x*pi)*cos(x)*cos(y) + 2*k*cos(x)*cos(y)'
    symbol_names = 'rho cp k'
    symbol_values = '${rho} ${cp} ${k}'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       NONZERO               mumps'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2u]
    type = ElementL2Error
    variable = u
    function = exact_u
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2v]
    variable = v
    function = exact_v
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2T]
    variable = T
    function = exact_T
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
  [L2p]
    variable = pressure
    function = exact_p
    type = ElementL2Error
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(modules/phase_field/tutorials/spinodal_decomposition/s5_energycurve.i)

#
# Example simulation of an iron-chromium alloy at 500 C. Equilibrium
# concentrations are at 23.6 and 82.3 mol% Cr. Kappa value, free energy equation,
# and mobility equation were provided by Lars Hoglund. Solved using the split
# form of the Cahn-Hilliard equation.

[Mesh]
  type = GeneratedMesh
  dim = 2
  elem_type = QUAD4
  nx = 25
  ny = 25
  nz = 0
  xmin = 0
  xmax = 25
  ymin = 0
  ymax = 25
  zmin = 0
  zmax = 0
  uniform_refine = 2
[]

[Variables]
  [./c]   # Mole fraction of Cr (unitless)
    order = FIRST
    family = LAGRANGE
    scaling = 1e+04
  [../]
  [./w]   # Chemical potential (eV/mol)
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./f_density]   # Local energy density (eV/mol)
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[ICs]
  [./concentrationIC]   # 46.774 mol% Cr with variations
    type = RandomIC
    min = 0.44774
    max = 0.48774
    seed = 210
    variable = c
  [../]
[]

[BCs]
  [./Periodic]
    [./c_bcs]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Kernels]
  [./w_dot]
    variable = w
    v = c
    type = CoupledTimeDerivative
  [../]
  [./coupled_res]
    variable = w
    type = SplitCHWRes
    mob_name = M
  [../]
  [./coupled_parsed]
    variable = c
    type = SplitCHParsed
    f_name = f_loc
    kappa_name = kappa_c
    w = w
  [../]
[]

[AuxKernels]
  # Calculates the energy density by combining the local and gradient energies
  [./f_density]   # (eV/mol/nm^2)
    type = TotalFreeEnergy
    variable = f_density
    f_name = 'f_loc'
    kappa_names = 'kappa_c'
    interfacial_vars = c
  [../]
[]

[Materials]
  # d is a scaling factor that makes it easier for the solution to converge
  # without changing the results. It is defined in each of the first three
  # materials and must have the same value in each one.
  [./kappa]                  # Gradient energy coefficient (eV nm^2/mol)
    type = GenericFunctionMaterial
    prop_names = 'kappa_c'
    prop_values = '8.125e-16*6.24150934e+18*1e+09^2*1e-27'
                  # kappa_c *eV_J*nm_m^2* d
  [../]
  [./mobility]               # Mobility (nm^2 mol/eV/s)
    # NOTE: This is a fitted equation, so only 'Conv' has units
    type = DerivativeParsedMaterial
    property_name = M
    coupled_variables = c
    constant_names =       'Acr    Bcr    Ccr    Dcr
                            Ecr    Fcr    Gcr
                            Afe    Bfe    Cfe    Dfe
                            Efe    Ffe    Gfe
                            nm_m   eV_J   d'
    constant_expressions = '-32.770969 -25.8186669 -3.29612744 17.669757
                            37.6197853 20.6941796  10.8095813
                            -31.687117 -26.0291774 0.2286581   24.3633544
                            44.3334237 8.72990497  20.956768
                            1e+09      6.24150934e+18          1e-27'
    expression = 'nm_m^2/eV_J/d*((1-c)^2*c*10^
                (Acr*c+Bcr*(1-c)+Ccr*c*log(c)+Dcr*(1-c)*log(1-c)+
                Ecr*c*(1-c)+Fcr*c*(1-c)*(2*c-1)+Gcr*c*(1-c)*(2*c-1)^2)
                +c^2*(1-c)*10^
                (Afe*c+Bfe*(1-c)+Cfe*c*log(c)+Dfe*(1-c)*log(1-c)+
                Efe*c*(1-c)+Ffe*c*(1-c)*(2*c-1)+Gfe*c*(1-c)*(2*c-1)^2))'
    derivative_order = 1
    outputs = exodus
  [../]
  [./local_energy]           # Local free energy function (eV/mol)
    type = DerivativeParsedMaterial
    property_name = f_loc
    coupled_variables = c
    constant_names = 'A   B   C   D   E   F   G  eV_J  d'
    constant_expressions = '-2.446831e+04 -2.827533e+04 4.167994e+03 7.052907e+03
                            1.208993e+04 2.568625e+03 -2.354293e+03
                            6.24150934e+18 1e-27'
    expression = 'eV_J*d*(A*c+B*(1-c)+C*c*log(c)+D*(1-c)*log(1-c)+
                E*c*(1-c)+F*c*(1-c)*(2*c-1)+G*c*(1-c)*(2*c-1)^2)'
    derivative_order = 2
  [../]
  [./precipitate_indicator]  # Returns 1/625 if precipitate
    type = ParsedMaterial
    property_name = prec_indic
    coupled_variables = c
    expression = if(c>0.6,0.0016,0)
  [../]
[]

[Postprocessors]
  [./step_size]             # Size of the time step
    type = TimestepSize
  [../]
  [./iterations]            # Number of iterations needed to converge timestep
    type = NumNonlinearIterations
  [../]
  [./nodes]                 # Number of nodes in mesh
    type = NumNodes
  [../]
  [./evaluations]           # Cumulative residual calculations for simulation
    type = NumResidualEvaluations
  [../]
  [./total_energy]          # Total free energy at each timestep
    type = ElementIntegralVariablePostprocessor
    variable = f_density
    execute_on = 'initial timestep_end'
  [../]
  [./num_features]          # Number of precipitates formed
    type = FeatureFloodCount
    variable = c
    threshold = 0.6
  [../]
  [./precipitate_area]      # Fraction of surface devoted to precipitates
    type = ElementIntegralMaterialProperty
    mat_prop = prec_indic
  [../]
  [./active_time]           # Time computer spent on simulation
    type = PerfGraphData
    section_name = "Root"
    data_type = total
  [../]
[]

[Preconditioning]
  [./coupled]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  l_max_its = 30
  l_tol = 1e-6
  nl_max_its = 50
  nl_abs_tol = 1e-9
  end_time = 604800   # 7 days
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_ksp_type
                         -sub_pc_type -pc_asm_overlap'
  petsc_options_value = 'asm      31                  preonly
                         ilu          1'
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    cutback_factor = 0.8
    growth_factor = 1.5
    optimal_iterations = 7
  [../]
  [./Adaptivity]
    coarsen_fraction = 0.1
    refine_fraction = 0.7
    max_h_level = 2
  [../]
[]

[Outputs]
  exodus = true
  console = true
  csv = true
  [./console]
    type = Console
    max_rows = 10
  [../]
[]

(modules/phase_field/test/tests/GrandPotentialPFM/SinteringParabolic.i)

#input file to test the GrandPotentialSinteringMaterial using the parabolic energy profile

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 17
  ny = 10
  xmin = 0
  xmax = 660
  ymin = 0
  ymax = 380
[]

[GlobalParams]
  op_num = 2
  var_name_base = gr
  int_width = 40
[]

[Variables]
  [./w]
    [./InitialCondition]
      type = FunctionIC
      variable = w
      function = f_w
    [../]
  [../]
  [./phi]
  [../]
  [./PolycrystalVariables]
  [../]
[]

[AuxVariables]
  [./T]
    order = CONSTANT
    family = MONOMIAL
    [./InitialCondition]
      type = FunctionIC
      variable = T
      function = f_T
    [../]
  [../]
[]

[ICs]
  [./phi_IC]
    type = SpecifiedSmoothCircleIC
    variable = phi
    x_positions = '190 470'
    y_positions = '190 190'
    z_positions = '  0   0'
    radii = '150 150'
    invalue = 0
    outvalue = 1
  [../]
  [./gr0_IC]
    type = SmoothCircleIC
    variable = gr0
    x1 = 190
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
  [./gr1_IC]
    type = SmoothCircleIC
    variable = gr1
    x1 = 470
    y1 = 190
    z1 = 0
    radius = 150
    invalue = 1
    outvalue = 0
  [../]
[]

[Functions]
  [./f_T]
    type = ConstantFunction
    value = 1600
  [../]
  [./f_w]
    type = ParsedFunction
    expression = '1.515e-7 * x'
  [../]
[]

[Materials]
  # Free energy coefficients for parabolic curve
  [./ks]
    type = ParsedMaterial
    property_name = ks
    coupled_variables = 'T'
    constant_names = 'a b'
    constant_expressions = '-0.0025 157.16'
    expression = 'a*T + b'
  [../]
  [./kv]
    type = ParsedMaterial
    property_name = kv
    material_property_names = 'ks'
    expression = '10 * ks'
  [../]
  # Diffusivity and mobilities
  [./chiD]
    type = GrandPotentialTensorMaterial
    f_name = chiD
    solid_mobility = L
    void_mobility = Lv
    chi = chi
    surface_energy = 19.7
    c = phi
    T = T
    D0 = 2.0e11
    GBmob0 = 1.4759e9
    Q = 2.77
    Em = 2.40
    bulkindex = 1
    gbindex = 20
    surfindex = 100
  [../]
  # Equilibrium vacancy concentration
  [./cs_eq]
    type = DerivativeParsedMaterial
    property_name = cs_eq
    coupled_variables = 'gr0 gr1 T'
    constant_names = 'Ef Egb kB'
    constant_expressions = '2.69 2.1 8.617343e-5'
    expression = 'bnds:=gr0^2 + gr1^2; cb:=exp(-Ef/kB/T); cgb:=exp(-(Ef-Egb)/kB/T);
                cb + 4.0*(cgb-cb)*(1.0 - bnds)^2'
  [../]
  # Everything else
  [./sintering]
    type = GrandPotentialSinteringMaterial
    chemical_potential = w
    void_op = phi
    Temperature = T
    surface_energy = 19.7
    grainboundary_energy = 9.86
    void_energy_coefficient = kv
    equilibrium_vacancy_concentration = cs_eq
    solid_energy_model = PARABOLIC
    outputs = exodus
  [../]

  # Concentration is only meant for output
  [./c]
    type = ParsedMaterial
    property_name = c
    material_property_names = 'hs rhos hv rhov'
    constant_names = 'Va'
    constant_expressions = '0.04092'
    expression = 'Va*(hs*rhos + hv*rhov)'
    outputs = exodus
  [../]
[]

[Kernels]
  [./dt_gr0]
    type = TimeDerivative
    variable = gr0
  [../]
  [./dt_gr1]
    type = TimeDerivative
    variable = gr1
  [../]
  [./dt_phi]
    type = TimeDerivative
    variable = phi
  [../]
  [./dt_w]
    type = TimeDerivative
    variable = w
  [../]
[]

[AuxKernels]
  [./T_aux]
    type = FunctionAux
    variable = T
    function = f_T
  [../]
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = JFNK
  dt = 1
  num_steps = 2
  nl_abs_tol = 1e-10
[]

[Outputs]
  exodus = true
[]

(modules/phase_field/examples/multiphase/GrandPotential3Phase_AD.i)

# This is an example of implementation of the multi-phase, multi-order parameter
# grand potential based phase-field model described in Phys. Rev. E, 98, 023309
# (2018). It includes 3 phases with 1 grain of each phase. This example was used
# to generate the results shown in Fig. 3 of the paper.

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 60
  xmin = -15
  xmax = 15
[]

[Variables]
  [w]
  []
  [etaa0]
  []
  [etab0]
  []
  [etad0]
  []
[]

[ICs]
  [IC_etaa0]
    type = FunctionIC
    variable = etaa0
    function = ic_func_etaa0
  []
  [IC_etab0]
    type = FunctionIC
    variable = etab0
    function = ic_func_etab0
  []
  [IC_etad0]
    type = ConstantIC
    variable = etad0
    value = 0.1
  []
  [IC_w]
    type = FunctionIC
    variable = w
    function = ic_func_w
  []
[]

[Functions]
  [ic_func_etaa0]
    type = ADParsedFunction
    value = '0.9*0.5*(1.0-tanh((x)/sqrt(2.0)))'
  []
  [ic_func_etab0]
    type = ADParsedFunction
    value = '0.9*0.5*(1.0+tanh((x)/sqrt(2.0)))'
  []
  [ic_func_w]
    type = ADParsedFunction
    value = 0
  []
[]

[Kernels]
# Order parameter eta_alpha0
  [ACa0_bulk]
    type = ADACGrGrMulti
    variable = etaa0
    v =           'etab0 etad0'
    gamma_names = 'gab   gad'
  []
  [ACa0_sw]
    type = ADACSwitching
    variable = etaa0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
  []
  [ACa0_int]
    type = ADACInterface
    variable = etaa0
    kappa_name = kappa
    variable_L = false
  []
  [ea0_dot]
    type = ADTimeDerivative
    variable = etaa0
  []
# Order parameter eta_beta0
  [ACb0_bulk]
    type = ADACGrGrMulti
    variable = etab0
    v =           'etaa0 etad0'
    gamma_names = 'gab   gbd'
  []
  [ACb0_sw]
    type = ADACSwitching
    variable = etab0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
  []
  [ACb0_int]
    type = ADACInterface
    variable = etab0
    kappa_name = kappa
    variable_L = false
  []
  [eb0_dot]
    type = ADTimeDerivative
    variable = etab0
  []
# Order parameter eta_delta0
  [ACd0_bulk]
    type = ADACGrGrMulti
    variable = etad0
    v =           'etaa0 etab0'
    gamma_names = 'gad   gbd'
  []
  [ACd0_sw]
    type = ADACSwitching
    variable = etad0
    Fj_names  = 'omegaa omegab omegad'
    hj_names  = 'ha     hb     hd'
  []
  [ACd0_int]
    type = ADACInterface
    variable = etad0
    kappa_name = kappa
    variable_L = false
  []
  [ed0_dot]
    type = ADTimeDerivative
    variable = etad0
  []
#Chemical potential
  [w_dot]
    type = ADSusceptibilityTimeDerivative
    variable = w
    f_name = chi
  []
  [Diffusion]
    type = ADMatDiffusion
    variable = w
    diffusivity = Dchi
  []
  [coupled_etaa0dot]
    type = ADCoupledSwitchingTimeDerivative
    variable = w
    v = etaa0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    args = 'etaa0 etab0 etad0'
  []
  [coupled_etab0dot]
    type = ADCoupledSwitchingTimeDerivative
    variable = w
    v = etab0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    args = 'etaa0 etab0 etad0'
  []
  [coupled_etad0dot]
    type = ADCoupledSwitchingTimeDerivative
    variable = w
    v = etad0
    Fj_names = 'rhoa rhob rhod'
    hj_names = 'ha   hb   hd'
    args = 'etaa0 etab0 etad0'
  []
[]

[Materials]
  [ha_test]
    type = ADSwitchingFunctionMultiPhaseMaterial
    h_name = ha
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etaa0'
  []
  [hb_test]
    type = ADSwitchingFunctionMultiPhaseMaterial
    h_name = hb
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etab0'
  []
  [hd_test]
    type = ADSwitchingFunctionMultiPhaseMaterial
    h_name = hd
    all_etas = 'etaa0 etab0 etad0'
    phase_etas = 'etad0'
  []
  [omegaa]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = omegaa
    material_property_names = 'Vm ka caeq'
    function = '-0.5*w^2/Vm^2/ka-w/Vm*caeq'
    derivative_order = 2
  []
  [omegab]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = omegab
    material_property_names = 'Vm kb cbeq'
    function = '-0.5*w^2/Vm^2/kb-w/Vm*cbeq'
    derivative_order = 2
  []
  [omegad]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = omegad
    material_property_names = 'Vm kd cdeq'
    function = '-0.5*w^2/Vm^2/kd-w/Vm*cdeq'
    derivative_order = 2
  []
  [rhoa]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = rhoa
    material_property_names = 'Vm ka caeq'
    function = 'w/Vm^2/ka + caeq/Vm'
    derivative_order = 2
  []
  [rhob]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = rhob
    material_property_names = 'Vm kb cbeq'
    function = 'w/Vm^2/kb + cbeq/Vm'
    derivative_order = 2
  []
  [rhod]
    type = ADDerivativeParsedMaterial
    args = 'w'
    f_name = rhod
    material_property_names = 'Vm kd cdeq'
    function = 'w/Vm^2/kd + cdeq/Vm'
    derivative_order = 2
  []
  [c]
    type = ADParsedMaterial
    material_property_names = 'Vm rhoa rhob rhod ha hb hd'
    function = 'Vm * (ha * rhoa + hb * rhob + hd * rhod)'
    f_name = c
  []
  [const]
    type = ADGenericConstantMaterial
    prop_names =  'kappa_c  kappa   L   D    Vm   ka    caeq kb    cbeq  kd    cdeq  gab gad gbd  mu  tgrad_corr_mult'
    prop_values = '0        1       1.0 1.0  1.0  10.0  0.1  10.0  0.9   10.0  0.5   1.5 1.5 1.5  1.0 0.0'
  []
  [Mobility]
    type = ADDerivativeParsedMaterial
    f_name = Dchi
    material_property_names = 'D chi'
    function = 'D*chi'
    derivative_order = 2
  []
  [chi]
    type = ADDerivativeParsedMaterial
    f_name = chi
    material_property_names = 'Vm ha(etaa0,etab0,etad0) ka hb(etaa0,etab0,etad0) kb hd(etaa0,etab0,etad0) kd'
    function = '(ha/ka + hb/kb + hd/kd) / Vm^2'
    args = 'etaa0 etab0 etad0'
    derivative_order = 2
  []
[]

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

[VectorPostprocessors]
  [etaa0]
    type = LineValueSampler
    variable = etaa0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  []
  [etab0]
    type = LineValueSampler
    variable = etab0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  []
  [etad0]
    type = LineValueSampler
    variable = etad0
    start_point = '-15 0 0'
    end_point = '15 0 0'
    num_points = 61
    sort_by = x
    execute_on = 'initial timestep_end final'
  []
[]

[Executioner]
  type = Transient
  nl_max_its = 15
  scheme = bdf2
  solve_type = NEWTON
  petsc_options_iname = -pc_type
  petsc_options_value = lu
  l_max_its = 15
  l_tol = 1.0e-3
  nl_rel_tol = 1.0e-8
  start_time = 0.0
  num_steps = 20
  nl_abs_tol = 1e-10
  dt = 1.0
[]

[Outputs]
  [exodus]
    type = Exodus
    execute_on = 'initial timestep_end final'
    interval = 1
  []
  [csv]
    type = CSV
    execute_on = 'initial timestep_end final'
    interval = 1
  []
[]

(modules/solid_mechanics/test/tests/ad_viscoplasticity_stress_update/gtn_single.i)

# This test provides an example of an individual GTN viscoplasticity model

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmax = 0.002
  ymax = 0.002
[]

[Physics/SolidMechanics/QuasiStatic/All]
  strain = FINITE
  add_variables = true
  base_name = 'total'
  generate_output = 'strain_xx strain_yy strain_xy hydrostatic_stress vonmises_stress'
  use_automatic_differentiation = true
[]

[Functions]
  [./pull]
    type = PiecewiseLinear
    x = '0 0.1'
    y = '0 1e-5'
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1e10
    poissons_ratio = 0.3
    base_name = 'total'
  [../]
  [./stress]
    type = ADComputeMultipleInelasticStress
    inelastic_models = gtn
    outputs = all
    base_name = 'total'
  [../]
  [./porosity]
    type = ADPorosityFromStrain
    initial_porosity = 0.1
    inelastic_strain = 'total_combined_inelastic_strain'
    outputs = 'all'
  [../]

  [./gtn]
    type = ADViscoplasticityStressUpdate
    total_strain_base_name = 'total'
    coefficient = 'coef'
    power = 3
    viscoplasticity_model = GTN
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./coef]
    type = ADParsedMaterial
    property_name = coef
    # Example of creep power law
    expression = '1e-18 * exp(-4e4 / 1.987 / 1200)'
  [../]
[]

[BCs]
  [./no_disp_x]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./no_disp_y]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./pull_disp_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = top
    function = pull
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 0.01
  end_time = 0.12
[]

[Postprocessors]
  [./disp_x]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  [../]
  [./disp_y]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
  [./avg_hydro]
    type = ElementAverageValue
    variable = total_hydrostatic_stress
  [../]
  [./avg_vonmises]
    type = ElementAverageValue
    variable = total_vonmises_stress
  [../]
  [./dt]
    type = TimestepSize
  [../]
  [./num_lin]
    type = NumLinearIterations
    outputs = console
  [../]
  [./num_nonlin]
    type = NumNonlinearIterations
    outputs = console
  [../]
  [./eff_creep_strain]
    type = ElementAverageValue
    variable = effective_viscoplasticity
  [../]
  [./porosity]
    type = ElementAverageValue
    variable = porosity
  [../]
[]

[Outputs]
  csv = true
[]

(modules/optimization/examples/diffusion_reaction/forward_exact.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 16
    ny = 16
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
  []
[]

[Variables/u]
[]

[Functions]
  [rxn_func]
    type = ParsedFunction
    expression = 'exp(x * y) - 1'
  []
[]

[Materials]
  [ad_dc_prop]
    type = ADParsedMaterial
    expression = '1 + u'
    coupled_variables = 'u'
    property_name = dc_prop
  []
  [ad_rxn_prop]
    type = ADGenericFunctionMaterial
    prop_values = 'rxn_func'
    prop_names = rxn_prop
    outputs = exodus
  []
  #ADMatReaction includes a negative sign in residual evaluation, so we need to
  #reverse this with a negative reaction rate. However, we wanted the parameter
  #to remain positive in the optimization problem so we do the same thing here,
  #which is why there is one object to evaluate function and another to flip
  #it's sign for the kernel
  [ad_neg_rxn_prop]
    type = ADParsedMaterial
    expression = '-rxn_prop'
    material_property_names = 'rxn_prop'
    property_name = 'neg_rxn_prop'
  []
[]

[Kernels]
  [udot]
    type = ADTimeDerivative
    variable = u
  []
  [diff]
    type = ADMatDiffusion
    variable = u
    diffusivity = dc_prop
  []
  [reaction]
    type = ADMatReaction
    variable = u
    reaction_rate = neg_rxn_prop
  []
  [src]
    type = ADBodyForce
    variable = u
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u
    boundary = 'left bottom'
    value = 0
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  dt = 0.1
  end_time = 1
[]

[Reporters]
  [sample]
    type = TransientSyntheticDataCreator
    variable = u
    single_set_of_measurement_points = '0.625 0.625 0  0.5 0.5 0  0.25 0.25 0
    0.875 0.625 0  0.625 0.875 0  0.75 0.25 0  0.25 0.75 0  0.8125 0.9375 0
    0.75   0.75   0
    0.8125 0.75   0
    0.875  0.75   0
    0.9375 0.75   0
    1      0.75   0
    0.8125 0.8125 0
    0.875  0.8125 0
    0.9375 0.8125 0
    1      0.8125 0
    0.875  0.875  0
    0.9375 0.875  0
    1      0.875  0
    0.9375 0.9375 0
    1      0.9375 0
    1      1      0'

    measurement_times_for_all_points = '0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0'
    outputs = csv_final
  []
[]

[AuxVariables]
  [reaction_rate]
  []
[]

[AuxKernels]
  [reaction_rate_aux]
    type = FunctionAux
    variable = reaction_rate
    function = rxn_func
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [u1]
    type = PointValue
    variable = u
    point = '0.25 0.25 0'
  []
  [u2]
    type = PointValue
    variable = u
    point = '0.75 0.75 0'
  []
  [u3]
    type = PointValue
    variable = u
    point = '1 1 0'
  []
[]

[Outputs]
  exodus = true
  csv = true
  [csv_final]
    type = CSV
    execute_on = 'FINAL'
    file_base = forward_exact_csv
  []
[]

(modules/combined/test/tests/optimization/compliance_sensitivity/paper_three_materials_test.i)

vol_frac = 0.4
cost_frac = 0.2 #0.283 # Change back to 0.4

power = 4

E0 = 1.0e-6
E1 = 0.2
E2 = 0.6
E3 = 1.0

rho0 = 1.0e-6
rho1 = 0.4
rho2 = 0.7
rho3 = 1.0

C0 = 1.0e-6
C1 = 0.5
C2 = 0.8
C3 = 1.0

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 50
    ny = 50
    xmin = 0
    xmax = 50
    ymin = 0
    ymax = 50
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push_left]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push_left
    coord = '25 0 0'
  []
  [push_center]
    type = ExtraNodesetGenerator
    input = push_left
    new_boundary = push_center
    coord = '50 0 0'
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]

[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [Cost]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]

[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]

[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []

[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-3
    mass = 1
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${E0}-${E1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${E0}-A1*${rho0}^${power}; E1:=A1*mat_den^${power}+B1; "
                 "A2:=(${E1}-${E2})/(${rho1}^${power}-${rho2}^${power}); "
                 "B2:=${E1}-A2*${rho1}^${power}; E2:=A2*mat_den^${power}+B2; "
                 "A3:=(${E2}-${E3})/(${rho2}^${power}-${rho3}^${power}); "
                 "B3:=${E2}-A3*${rho2}^${power}; E3:=A3*mat_den^${power}+B3; "
                 "if(mat_den<${rho1},E1,if(mat_den<${rho2},E2,E3))"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []

  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^(1/${power})-${rho1}^(1/${power})); "
                 "B1:=${C0}-A1*${rho0}^(1/${power}); C1:=A1*mat_den^(1/${power})+B1; "
                 "A2:=(${C1}-${C2})/(${rho1}^(1/${power})-${rho2}^(1/${power})); "
                 "B2:=${C1}-A2*${rho1}^(1/${power}); C2:=A2*mat_den^(1/${power})+B2; "
                 "A3:=(${C2}-${C3})/(${rho2}^(1/${power})-${rho3}^(1/${power})); "
                 "B3:=${C2}-A3*${rho2}^(1/${power}); C3:=A3*mat_den^(1/${power})+B3; "
                 "if(mat_den<${rho1},C1,if(mat_den<${rho2},C2,C3))"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
  []
  [CostDensity]
    type = ParsedMaterial
    property_name = CostDensity
    coupled_variables = 'mat_den Cost'
    expression = 'mat_den*Cost'
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = Cost_mat
    outputs = 'exodus'
  []
[]

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

[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 2
    weights = linear
    prop_name = cost_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdateTwoConstraints
    # This is
    density_sensitivity = Dc
    cost_density_sensitivity = Cc
    cost = Cost
    cost_fraction = ${cost_frac}
    design_density = mat_den
    volume_fraction = ${vol_frac}

    bisection_lower_bound = 0
    bisection_upper_bound = 1.0e16 # 100

    relative_tolerance = 1.0e-3
    bisection_move = 0.02
    execute_on = TIMESTEP_BEGIN
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  # Provides Cc
  [calc_sense_cost]
    type = SensitivityFilter
    density_sensitivity = Cc
    design_density = mat_den
    filter_UO = rad_avg_cost
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu superlu_dist'
  nl_abs_tol = 1e-10
  dt = 1.0
  num_steps = 25
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]

[Postprocessors]
  [mesh_volume]
    type = VolumePostprocessor
    execute_on = 'initial timestep_end'
  []
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [vol_frac]
    type = ParsedPostprocessor
    expression = 'total_vol / mesh_volume'
    pp_names = 'total_vol mesh_volume'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralMaterialProperty
    mat_prop = CostDensity
  []
  [cost_frac]
    type = ParsedPostprocessor
    expression = 'cost / mesh_volume'
    pp_names = 'cost mesh_volume'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

(modules/solid_mechanics/test/tests/ad_viscoplasticity_stress_update/lps_single_split.i)

# This test provides an example of combining two LPS viscoplasticity model.
# The answer should be close, but not exactly the same, as lps_single.i

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmax = 0.002
  ymax = 0.002
[]

[Physics/SolidMechanics/QuasiStatic/All]
  strain = FINITE
  add_variables = true
  generate_output = 'strain_xx strain_yy strain_xy hydrostatic_stress vonmises_stress'
  use_automatic_differentiation = true
[]

[Functions]
  [./pull]
    type = PiecewiseLinear
    x = '0 0.1'
    y = '0 1e-5'
  [../]
  [./tot_effective_viscoplasticity]
    type = ParsedFunction
    symbol_values = 'lps_1_eff_creep_strain lps_2_eff_creep_strain'
    symbol_names = 'lps_1_eff_creep_strain lps_2_eff_creep_strain'
    expression = 'lps_1_eff_creep_strain+lps_2_eff_creep_strain'
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1e10
    poissons_ratio = 0.3
  [../]
  [./stress]
    type = ADComputeMultipleInelasticStress
    inelastic_models = 'one two'
    outputs = all
  [../]
  [./porosity]
    type = ADPorosityFromStrain
    initial_porosity = 0.1
    inelastic_strain = 'combined_inelastic_strain'
    outputs = 'all'
  [../]

  [./one]
    type = ADViscoplasticityStressUpdate
    coefficient = 'coef'
    power = 3
    base_name = 'lps_first'
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./two]
    type = ADViscoplasticityStressUpdate
    coefficient = 'coef'
    power = 3
    base_name = 'lps_second'
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./coef]
    type = ADParsedMaterial
    property_name = coef
    # Example of creep power law
    expression = '0.5e-18 * exp(-4e4 / 1.987 / 1200)'
  [../]
[]

[BCs]
  [./no_disp_x]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./no_disp_y]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./pull_disp_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = top
    function = pull
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 0.01
  end_time = 0.12
[]

[Postprocessors]
  [./disp_x]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  [../]
  [./disp_y]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
  [./avg_hydro]
    type = ElementAverageValue
    variable = hydrostatic_stress
  [../]
  [./avg_vonmises]
    type = ElementAverageValue
    variable = vonmises_stress
  [../]
  [./dt]
    type = TimestepSize
  [../]
  [./num_lin]
    type = NumLinearIterations
    outputs = console
  [../]
  [./num_nonlin]
    type = NumNonlinearIterations
    outputs = console
  [../]
  [./lps_1_eff_creep_strain]
    type = ElementAverageValue
    variable = lps_first_effective_viscoplasticity
    outputs = none
  [../]
  [./lps_2_eff_creep_strain]
    type = ElementAverageValue
    variable = lps_second_effective_viscoplasticity
    outputs = none
  [../]
  [./eff_creep_strain_tot]
    type = FunctionValuePostprocessor
    function = tot_effective_viscoplasticity
  [../]
  [./porosity]
    type = ElementAverageValue
    variable = porosity
  [../]
[]

[Outputs]
  csv = true
[]

(tutorials/tutorial02_multiapps/step03_coupling/03_parent_subcycling_picard.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
  []
[]

[AuxVariables]
  [vt]
  []
[]

[Kernels]
  [diff]
    type = MatDiffusion
    variable = u
  []
  [force]
    type = BodyForce
    variable = u
    value = 1.
  []
  [td]
    type = TimeDerivative
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Materials]
  [diff]
    type = ParsedMaterial
    property_name = D
    coupled_variables = 'vt'
    expression = 'vt'
  []
[]

[Executioner]
  type = Transient
  end_time = 2
  dt = 0.2

  fixed_point_max_its = 10

  nl_abs_tol = 1e-10
  fixed_point_rel_tol = 1e-6
  fixed_point_abs_tol = 1e-10

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

[MultiApps]
  [micro]
    type = TransientMultiApp
    positions = '0.15 0.15 0  0.45 0.45 0  0.75 0.75 0'
    input_files = '03_sub_subcycling_picard.i'
    execute_on = timestep_end
    output_in_position = true
    sub_cycling = true
  []
[]

[Transfers]
  [push_u]
    type = MultiAppVariableValueSampleTransfer
    to_multi_app = micro
    source_variable = u
    variable = ut
  []

  [pull_v]
    type = MultiAppPostprocessorInterpolationTransfer
    from_multi_app = micro
    variable = vt
    postprocessor = average_v
  []
[]

(modules/solid_mechanics/test/tests/ad_viscoplasticity_stress_update/lps_dual.i)

# This test provides an example of combining two LPS viscoplasticity models with different stress
# exponents.

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmax = 0.002
  ymax = 0.002
[]

[Variables]
  [./temp]
    initial_condition = 1000
  [../]
[]

[Kernels]
  [./dt]
    type = ADTimeDerivative
    variable = temp
  [../]
  [./diff]
    type = ADDiffusion
    variable = temp
  [../]
[]

[Physics/SolidMechanics/QuasiStatic/All]
  strain = FINITE
  add_variables = true
  generate_output = 'strain_xx strain_yy strain_xy hydrostatic_stress vonmises_stress'
  use_automatic_differentiation = true
[]

[Functions]
  [./pull]
    type = PiecewiseLinear
    x = '0 0.1'
    y = '0 1e-5'
  [../]
  [./tot_effective_viscoplasticity]
    type = ParsedFunction
    symbol_values = 'lps_1_eff_creep_strain lps_3_eff_creep_strain'
    symbol_names = 'lps_1_eff_creep_strain lps_3_eff_creep_strain'
    expression = 'lps_1_eff_creep_strain+lps_3_eff_creep_strain'
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1e10
    poissons_ratio = 0.3
  [../]
  [./stress]
    type = ADComputeMultipleInelasticStress
    inelastic_models = 'one two'
    outputs = all
  [../]
  [./porosity]
    type = ADPorosityFromStrain
    initial_porosity = 0.1
    inelastic_strain = 'combined_inelastic_strain'
    outputs = 'all'
  [../]

  [./one]
    type = ADViscoplasticityStressUpdate
    coefficient = 'coef_3'
    power = 3
    base_name = 'lps_1'
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./two]
    type = ADViscoplasticityStressUpdate
    coefficient = 1e-10
    power = 1
    base_name = 'lps_3'
    outputs = all
    relative_tolerance = 1e-11
  [../]
  [./coef]
    type = ADParsedMaterial
    property_name = coef_3
    # Example of creep power law
    coupled_variables = temp
    expression = '0.5e-18 * exp(-4e4 / 1.987 / temp)'
  [../]
[]

[BCs]
  [./no_disp_x]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./no_disp_y]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./pull_disp_y]
    type = ADFunctionDirichletBC
    variable = disp_y
    boundary = top
    function = pull
  [../]
  [./temp_ramp]
    type = ADFunctionDirichletBC
    boundary = right
    function = '1000 + 400 * t / 0.12'
    variable = temp
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 0.01
  end_time = 0.12
[]

[Postprocessors]
  [./disp_x]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  [../]
  [./disp_y]
    type = SideAverageValue
    variable = disp_y
    boundary = top
  [../]
  [./avg_hydro]
    type = ElementAverageValue
    variable = hydrostatic_stress
  [../]
  [./avg_vonmises]
    type = ElementAverageValue
    variable = vonmises_stress
  [../]
  [./dt]
    type = TimestepSize
  [../]
  [./num_lin]
    type = NumLinearIterations
    outputs = console
  [../]
  [./num_nonlin]
    type = NumNonlinearIterations
    outputs = console
  [../]
  [./lps_1_eff_creep_strain]
    type = ElementAverageValue
    variable = lps_1_effective_viscoplasticity
  [../]
  [./lps_3_eff_creep_strain]
    type = ElementAverageValue
    variable = lps_3_effective_viscoplasticity
  [../]
  [./lps_1_gauge_stress]
    type = ElementAverageValue
    variable = lps_1_gauge_stress
  [../]
  [./lps_3_gauge_stress]
    type = ElementAverageValue
    variable = lps_3_gauge_stress
  [../]
  [./eff_creep_strain_tot]
    type = FunctionValuePostprocessor
    function = tot_effective_viscoplasticity
  [../]
  [./porosity]
    type = ElementAverageValue
    variable = porosity
  [../]
[]

[Outputs]
  csv = true
[]