- prop_nameName of the material property to average
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of the material property to average
 - radiusCut-off radius for the averaging
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Cut-off radius for the averaging
 - weightslinearDistance based weight function
Default:linear
C++ Type:MooseEnum
Controllable:No
Description:Distance based weight function
 
RadialAverage
Perform a radial average of a material property
Given a material property and a radius for averaging, the RadialAverage object computes the spatial average value of the property over the radius.
Applications
This can be used for nonlocal damage models in the SolidMechanics module where the damage_index that is used for computing the damage stress is average over a certain radius . This can help alleviate mesh sensitivity in certain cases. This can be accomplished by running the RadialAverage object on a local damage material property. Then using the NonlocalDamage model in conjunction with the ComputeDamageStress the damage index used for updating the stress is averaged over a certain radius.
Design
The RadialAverage user object is derived from ElementUserObject and works in two stages.
In the element loop (in the
execute()method) a list of all quadrature points, their locations, indices, and selected material property value is compiled.In the
finalize()method1. the list is communicated in parallel
2. a KD-tree is filled with all quadrature point entries (utilizing the nanoflann library bundled with libMesh)
3. a loop over all QPs is performed and at each QP a (
radius) radius search in the KD-tree is performed4. the results from the search are used to spatially averaged
The "weights" parameter determines the distance based weight function to be used in the averaging process. constant assigns an equal weight to all material points, linear weights each point by  (a linear fall-off with distance), and cosine weights each point by .
The weights mentioned above currently do not take into account the quadrature point weight.
Input Parameters
- blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
 
Optional Parameters
- allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Controllable:No
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
 - execute_onTIMESTEP_BEGINThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:TIMESTEP_BEGIN
C++ Type:ExecFlagEnum
Options:XFEM_MARK, FORWARD, ADJOINT, HOMOGENEOUS_FORWARD, ADJOINT_TIMESTEP_BEGIN, ADJOINT_TIMESTEP_END, NONE, INITIAL, LINEAR, LINEAR_CONVERGENCE, NONLINEAR, NONLINEAR_CONVERGENCE, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, MULTIAPP_FIXED_POINT_CONVERGENCE, FINAL, CUSTOM, TRANSFER
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
 - execution_order_group0Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
Default:0
C++ Type:int
Controllable:No
Description:Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
 - force_postauxFalseForces the UserObject to be executed in POSTAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in POSTAUX
 - force_preauxFalseForces the UserObject to be executed in PREAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREAUX
 - force_preicFalseForces the UserObject to be executed in PREIC during initial setup
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREIC during initial setup
 
Execution Scheduling Parameters
- control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
 - enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
 - implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
 - padding0Padding for communication. This gets added to the radius when determining which QPs to send to other processors. Increase this gradually if inconsistent parallel results occur.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Padding for communication. This gets added to the radius when determining which QPs to send to other processors. Increase this gradually if inconsistent parallel results occur.
 - seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
 - use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
 
Advanced Parameters
- prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
 - use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
 
Material Property Retrieval Parameters
Input Files
- (modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material_cost.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material_cost_initial.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/paper_three_materials_test.i)
 - (modules/combined/examples/optimization/multi-load/single_subapp_two.i)
 - (modules/combined/examples/optimization/multi-load/square_subapp_one.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/2d_mbb.i)
 - (modules/combined/examples/optimization/multi-load/single_subapp_one.i)
 - (modules/combined/test/tests/optimization/optimization_density_update/top_opt_3d.i)
 - (test/tests/userobjects/radial_average/test.i)
 - (modules/optimization/test/tests/simp/2d_twoconstraints.i)
 - (modules/combined/examples/optimization/multi-load/square_subapp_two.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/three_materials_thermal.i)
 - (modules/combined/examples/optimization/thermomechanical/thermal_sub.i)
 - (modules/combined/test/tests/optimization/thermal_sensitivity/2d_root.i)
 - (modules/solid_mechanics/test/tests/scalar_material_damage/ad_nonlocal_scalar_damage.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/thermal_test.i)
 - (modules/solid_mechanics/test/tests/scalar_material_damage/nonlocal_scalar_damage.i)
 - (modules/combined/examples/optimization/2d_mbb.i)
 - (modules/combined/test/tests/optimization/optimization_density_update/top_opt_2d.i)
 - (modules/optimization/test/tests/simp/2d.i)
 - (modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material.i)
 - (modules/combined/examples/optimization/thermomechanical/structural_sub.i)
 - (modules/combined/examples/optimization/three_materials.i)
 
References
No citations exist within this document.weights
Default:linear
C++ Type:MooseEnum
Options:constant, linear, cosine
Controllable:No
Description:Distance based weight function
(modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material_cost.i)
vol_frac = 0.5
power = 3
E0 = 1.0e-6
E1 = 0.3
E2 = 1.0
rho0 = 1.0e-6
rho1 = 0.3
rho2 = 1.0
C0 = 1.0e-6
C1 = 0.5
C2 = 1.0
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 150
    ny = 50
    xmin = 0
    xmax = 30
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push
    coord = '30 10 0'
  []
[]
[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push]
    type = NodalGravity
    variable = disp_y
    boundary = push
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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^${power}+B2; "
                 "if(mat_den<${rho1},E1,E2)"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [Cost]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${C0}-${C1})/(${rho0}^${power}-${rho1}^${power}); "
                 "B1:=${C0}-A1*${rho0}^${power}; C1:=A1*mat_den^${power}+B1; "
                 "A2:=(${C1}-${C2})/(${rho1}^${power}-${rho2}^${power}); "
                 "B2:=${C1}-A2*${rho1}^${power}; C2:=A2*mat_den^${power}+B2; "
                 "if(mat_den<${rho1},C1,C2)"
    coupled_variables = 'mat_den'
    property_name = 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
    outputs = 'exodus'
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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-8
  dt = 1.0
  num_steps = 70
[]
[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'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
[]
(modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material_cost_initial.i)
vol_frac = 0.4
cost_frac = 0.22 # Change back to 0.4
power = 2.0
E0 = 1.0e-6
E1 = 0.3
E2 = 1.0
rho0 = 1.0e-6
rho1 = 0.3
rho2 = 1.0
C0 = 1.0e-6
C1 = 0.5
C2 = 1.0
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 150
    ny = 50
    xmin = 0
    xmax = 30
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push
    coord = '30 10 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 = ${vol_frac}
  []
[]
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push]
    type = NodalGravity
    variable = disp_y
    boundary = push
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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^${power}+B2; "
                 "if(mat_den<${rho1},E1,E2)"
    coupled_variables = 'mat_den'
    property_name = E_phys
    epsilon = 1e-12
  []
  [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; "
                 "if(mat_den<${rho1},C1,C2)"
    coupled_variables = 'mat_den'
    property_name = Cost_mat
    epsilon = 1e-12
  []
  [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
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 1.2
    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
    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-8
  dt = 1.0
  num_steps = 10 #50000
[]
[Outputs]
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]
[Postprocessors]
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
  [cost_sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = cost_sensitivity
  []
  [cost]
    type = ElementIntegralVariablePostprocessor
    variable = Cost
  []
[]
(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 = ${vol_frac}
  []
[]
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-3
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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<${rho2},E2,E3))"
    coupled_variables = 'mat_den'
    property_name = E_phys
    epsilon = 1e-12
  []
  [Cost_mat]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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
    epsilon = 1e-12
  []
  [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
  []
[]
[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]
  [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/combined/examples/optimization/multi-load/single_subapp_two.i)
power = 2
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [Bottom]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 80
    ny = 40
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 75
  []
  [left_load]
    type = ExtraNodesetGenerator
    input = Bottom
    new_boundary = left_load
    coord = '37.5 75 0'
  []
  [right_load]
    type = ExtraNodesetGenerator
    input = left_load
    new_boundary = right_load
    coord = '112.5 75 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]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.1
  []
  [sensitivity_var]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
[]
[AuxKernels]
  [sensitivity_kernel]
    type = MaterialRealAux
    check_boundary_restricted = false
    property = sensitivity
    variable = sensitivity_var
    execute_on = 'TIMESTEP_END'
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [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
  []
[]
[NodalKernels]
  [push_right]
    type = NodalGravity
    variable = disp_y
    boundary = right_load
    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
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.0
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
[]
[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
  []
  # No SIMP optimization in subapp
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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
    execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(modules/combined/examples/optimization/multi-load/square_subapp_one.i)
power = 1.0
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [Bottom]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 100
    ny = 100
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 150
  []
  [left_load]
    type = ExtraNodesetGenerator
    input = Bottom
    new_boundary = left_load
    coord = '0 150 0'
  []
  [right_load]
    type = ExtraNodesetGenerator
    input = left_load
    new_boundary = right_load
    coord = '150 150 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]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.25
  []
  [sensitivity_var]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
[]
[AuxKernels]
  [sensitivity_kernel]
    type = MaterialRealAux
    property = sensitivity
    variable = sensitivity_var
    check_boundary_restricted = false
    execute_on = 'TIMESTEP_END'
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [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
  []
  [no_x_right]
    type = DirichletBC
    variable = disp_x
    boundary = right_support
    value = 0.0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = left_load
    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
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  # We do averaging in subapps
  [rad_avg]
    type = RadialAverage
    radius = 8
    weights = linear
    prop_name = sensitivity
    force_preaux = true
    execute_on = 'TIMESTEP_END'
  []
  # Provides Dc
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    force_postaux = true
    execute_on = 'TIMESTEP_END'
  []
[]
[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 = 10
[]
[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
    execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(modules/combined/test/tests/optimization/compliance_sensitivity/2d_mbb.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-8
power = 2
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 150
    ny = 50
    xmin = 0
    xmax = 30
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push
    coord = '30 10 0'
  []
[]
[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push]
    type = NodalGravity
    variable = disp_y
    boundary = push
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
    incremental = false
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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-8
  dt = 1.0
  num_steps = 70
[]
[Outputs]
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]
[Postprocessors]
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
[]
(modules/combined/examples/optimization/multi-load/single_subapp_one.i)
power = 2
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  # final_generator = 'MoveRight'
  [Bottom]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 80
    ny = 40
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 75
  []
  [left_load]
    type = ExtraNodesetGenerator
    input = Bottom
    new_boundary = left_load
    coord = '37.5 75 0'
  []
  [right_load]
    type = ExtraNodesetGenerator
    input = left_load
    new_boundary = right_load
    coord = '112.5 75 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]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.1
  []
  [sensitivity_var]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
[]
[AuxKernels]
  [sensitivity_kernel]
    type = MaterialRealAux
    property = sensitivity
    variable = sensitivity_var
    check_boundary_restricted = false
    execute_on = 'TIMESTEP_END'
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [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
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = left_load
    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
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.0
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity
    design_density = mat_den
    youngs_modulus = E_phys
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 3
    weights = linear
    prop_name = sensitivity
    force_preaux = true
    execute_on = 'TIMESTEP_END'
  []
  # No SIMP optimization in subapp
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    force_postaux = true
    execute_on = 'TIMESTEP_END'
  []
[]
[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
    execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(modules/combined/test/tests/optimization/optimization_density_update/top_opt_3d.i)
vol_frac = 0.5
E0 = 1e5
Emin = 1e-2
power = 2
[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 24
    ny = 12
    nz = 12
    xmin = 0
    xmax = 20
    ymin = 0
    ymax = 10
    zmin = 0
    zmax = 10
  []
  [middle_bottom_left_edge]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = pull
    coord = '0 0 5'
  []
[]
[AuxVariables]
  [compliance]
    family = MONOMIAL
    order = CONSTANT
  []
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = right
    value = 0.0
  []
  [no_z]
    type = DirichletBC
    variable = disp_z
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [pull]
    type = NodalGravity
    variable = disp_y
    boundary = pull
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
    incremental = false
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.5
    weights = constant
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    execution_order_group = -1
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
  []
[]
[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu '
  nl_abs_tol = 1e-10
  l_max_its = 200
  start_time = 0.0
  dt = 1.0
  num_steps = 10
[]
[Outputs]
  [out]
    type = Exodus
    time_step_interval = 10
  []
[]
(test/tests/userobjects/radial_average/test.i)
[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = -1
    xmax = 1
    ymin = -1
    ymax = 1
    nx = 10
    ny = 10
  []
[]
[Problem]
  solve = false
  kernel_coverage_check = false
[]
[AuxVariables]
  [non_local_material]
    family = MONOMIAL
    order = SECOND
  []
[]
[AuxKernels]
  [non_local]
    type = RadialAverageAux
    average_UO = average
    variable = non_local_material
    execute_on = 'INITIAL TIMESTEP_BEGIN'
  []
[]
[Functions]
  [func]
    type = ParsedFunction
    expression = 'if(x >=0,(1+t),-(1+t))'
  []
[]
[Materials]
  [local_material]
    type = GenericFunctionMaterial
    prop_names = local
    prop_values = func
    outputs = exodus
  []
[]
[UserObjects]
  [average]
    type = RadialAverage
    prop_name = local
    weights = constant
    execute_on = "INITIAL timestep_end"
    radius = 0.3
  []
[]
[Executioner]
  type = Transient
  end_time = 3
  dt = 1
[]
[Outputs]
  exodus = true
[]
(modules/optimization/test/tests/simp/2d_twoconstraints.i)
cost_frac = 0.3
vol_frac = 0.2
[Mesh]
  [planet]
    type = ConcentricCircleMeshGenerator
    has_outer_square = false
    radii = 1
    num_sectors = 10
    rings = 2
    preserve_volumes = false
  []
  [moon]
    type = ConcentricCircleMeshGenerator
    has_outer_square = false
    radii = 0.5
    num_sectors = 8
    rings = 2
    preserve_volumes = false
  []
  [combine]
    type = CombinerGenerator
    inputs = 'planet moon'
    positions = '0 0 0 -1.5 -0.5 0'
  []
[]
[AuxVariables]
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.1
  []
  [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
  []
[]
[Variables]
  [u]
  []
  [v]
  []
[]
[Kernels]
  [diff_u]
    type = Diffusion
    variable = u
  []
  [dt_u]
    type = TimeDerivative
    variable = u
  []
  [diff_v]
    type = Diffusion
    variable = v
  []
  [dt_v]
    type = TimeDerivative
    variable = v
  []
[]
[Materials]
  [thermal_cond]
    type = GenericFunctionMaterial
    prop_values = '-1.4*abs(y)-2.7*abs(x)'
    prop_names = thermal_cond
    outputs = 'exodus'
  []
  [thermal_compliance_sensitivity]
    type = GenericFunctionMaterial
    prop_values = '-3*abs(y)-1.5*abs(x)'
    prop_names = thermal_sensitivity
    outputs = 'exodus'
  []
  [cost_sensitivity]
    type = GenericFunctionMaterial
    prop_values = '-0.3*y*y-0.5*abs(x*y)'
    prop_names = cost_sensitivity
    outputs = 'exodus'
  []
  [cost_sensitivity_parsed]
    type = DerivativeParsedMaterial
    expression = "if(mat_den<0.2,1.0,0.5)"
    coupled_variables = 'mat_den'
    property_name = cost_sensitivity_parsed
  []
  [cc]
    type = CostSensitivity
    design_density = mat_den
    cost = cost_sensitivity_parsed
    outputs = 'exodus'
    declare_suffix = 'for_testing'
  []
[]
[BCs]
  [flux_u]
    type = DirichletBC
    variable = u
    boundary = outer
    value = 3.0
  []
  [flux_v]
    type = DirichletBC
    variable = v
    boundary = outer
    value = 7.0
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [rad_avg_cost]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = cost_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 = 1.0e16
    relative_tolerance = 1.0e-3
    bisection_move = 0.15
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
  [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
  dt = 0.1
  num_steps = 3
  nl_rel_tol = 1e-04
[]
[Outputs]
  exodus = true
[]
(modules/combined/examples/optimization/multi-load/square_subapp_two.i)
power = 1.0
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [Bottom]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 100
    ny = 100
    xmin = 0
    xmax = 150
    ymin = 0
    ymax = 150
  []
  [left_load]
    type = ExtraNodesetGenerator
    input = Bottom
    new_boundary = left_load
    coord = '0 150 0'
  []
  [right_load]
    type = ExtraNodesetGenerator
    input = left_load
    new_boundary = right_load
    coord = '150 150 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]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
  [Cc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.25
  []
  [sensitivity_var]
    family = MONOMIAL
    order = SECOND
    initial_condition = -1.0
  []
[]
[AuxKernels]
  [sensitivity_kernel]
    type = MaterialRealAux
    check_boundary_restricted = false
    property = sensitivity
    variable = sensitivity_var
    execute_on = 'TIMESTEP_END'
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [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
  []
  [no_x_right]
    type = DirichletBC
    variable = disp_x
    boundary = right_support
    value = 0.0
  []
[]
[NodalKernels]
  [push_right]
    type = NodalGravity
    variable = disp_y
    boundary = right_load
    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
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  # We do averaging in subapps
  [rad_avg]
    type = RadialAverage
    radius = 8
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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 = 10
[]
[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
    execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(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 = ${vol_frac}
  []
  [thermal_compliance]
    order = CONSTANT
    family = MONOMIAL
  []
[]
# [ICs]
#   [mat_den]
#     type = RandomIC
#     seed = 4
#     variable = mat_den
#     max = '${fparse vol_frac+0.25}'
#     min = '${fparse vol_frac-0.25}'
#   []
# []
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
  [thermal_compliance]
    type = MaterialRealAux
    property = thermal_compliance
    variable = thermal_compliance
    execute_on = timestep_end
  []
[]
[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [top]
    type = DirichletBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = DirichletBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = DirichletBC
    variable = temp
    boundary = left
    value = 0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-6 # -3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-6 # -3
    mass = 1
  []
[]
[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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<${rho2},TC2,TC3))"
    coupled_variables = 'mat_den'
    property_name = thermal_cond
  []
  [thermal_compliance]
    type = ThermalCompliance
    temperature = temp
    thermal_conductivity = thermal_cond
  []
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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
  []
  [tc]
    type = ThermalSensitivity
    design_density = mat_den
    thermal_conductivity = thermal_cond
    temperature = temp
  []
[]
[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]
  [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/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
    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 = ${vol_frac}
  []
[]
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]
[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [left_n1]
    type = DirichletBC
    variable = temp
    boundary = n1
    value = 0.0
  []
  [top]
    type = NeumannBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = NeumannBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = NeumannBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = NeumannBC
    variable = temp
    boundary = left
    value = 0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
[]
[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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/combined/test/tests/optimization/thermal_sensitivity/2d_root.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-4
power = 1
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    ny = 20
    xmin = 0
    xmax = 40
    ymin = 0
    ymax = 40
  []
[]
[Variables]
  [T]
    initial_condition = 100
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Kernels]
  [heat]
    type = HeatConduction
    diffusion_coefficient = k
    variable = T
  []
  [heat_source]
    type = HeatSource
    function = 1e-2
    variable = T
  []
[]
[DiracKernels]
  [src]
    type = ConstantPointSource
    variable = T
    point = '0 5 0'
    value = 10
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = T
    boundary = 'right top bottom'
    value = 0.0
  []
[]
[Materials]
  [k]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = k
  []
  [dc]
    type = ThermalSensitivity
    temperature = T
    design_density = mat_den
    thermal_conductivity = k
  []
  #only needed for objective function output in postprocessor
  [thermal_compliance]
    type = ThermalCompliance
    temperature = T
    thermal_conductivity = k
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 3
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = 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-8
  dt = 1.0
  dtmin = 1.0
  num_steps = 20
[]
[Outputs]
  [out]
    type = CSV
    execute_on = 'FINAL'
  []
  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 = thermal_sensitivity
  []
  [objective_thermal]
    type = ElementIntegralMaterialProperty
    mat_prop = thermal_compliance
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(modules/solid_mechanics/test/tests/scalar_material_damage/ad_nonlocal_scalar_damage.i)
[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]
[Mesh]
  type = GeneratedMesh
  dim = 3
  xmin = -0.5
  xmax = 0.5
  nx = 5
  ny = 5
  nz = 5
  elem_type = HEX8
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    incremental = true
    add_variables = true
    generate_output = 'stress_xx strain_xx'
    use_automatic_differentiation = true
  []
[]
[BCs]
  [symmy]
    type = ADDirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
  [symmx]
    type = ADDirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [symmz]
    type = ADDirichletBC
    variable = disp_z
    boundary = back
    value = 0
  []
  [axial_load]
    type = ADDirichletBC
    variable = disp_x
    boundary = right
    value = 0.01
  []
[]
[Functions]
  [func]
    type = ParsedFunction
    expression = 'if(x>=0,0.5*t, t)'
  []
[]
[UserObjects]
  [ele_avg]
    type = RadialAverage
    prop_name = local_damage_reg
    weights = constant
    execute_on = "INITIAL timestep_end"
    radius = 0.55
  []
[]
[Materials]
  [non_ad_local_damage]
    type = MaterialADConverter
    ad_props_in = local_damage
    reg_props_out = local_damage_reg
  []
  [local_damage_index]
    type = ADGenericFunctionMaterial
    prop_names = local_damage_index
    prop_values = func
  []
  [local_damage]
    type = ADScalarMaterialDamage
    damage_index = local_damage_index
    damage_index_name = local_damage
  []
  [damage]
    type = ADNonlocalDamage
    average_UO = ele_avg
    local_damage_model = local_damage
    damage_index_name = nonlocal_damage
  []
  [elasticity]
    type = ADComputeIsotropicElasticityTensor
    poissons_ratio = 0.2
    youngs_modulus = 10e9
  []
  [stress]
    type = ADComputeDamageStress
    damage_model = damage
  []
[]
[Postprocessors]
  [stress_xx]
    type = ElementAverageValue
    variable = stress_xx
  []
  [strain_xx]
    type = ElementAverageValue
    variable = strain_xx
  []
  [nonlocal_damage]
    type = ADElementAverageMaterialProperty
    mat_prop = nonlocal_damage
  []
  [local_damage]
    type = ADElementAverageMaterialProperty
    mat_prop = local_damage
  []
[]
[Executioner]
  type = Transient
  l_max_its = 50
  l_tol = 1e-8
  nl_max_its = 20
  nl_rel_tol = 1e-12
  nl_abs_tol = 1e-8
  dt = 0.2
  dtmin = 0.1
  end_time = 1
[]
[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 = ${vol_frac}
  []
[]
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]
[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
    diffusion_coefficient = thermal_cond
  []
  [heat_source]
    type = HeatSource
    value = 1e-2 # W/m^3
    variable = temp
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [left_n1]
    type = DirichletBC
    variable = temp
    boundary = n1
    value = 0.0
  []
  [top]
    type = NeumannBC
    variable = temp
    boundary = top
    value = 0
  []
  [bottom]
    type = NeumannBC
    variable = temp
    boundary = bottom
    value = 0
  []
  [right]
    type = NeumannBC
    variable = temp
    boundary = right
    value = 0
  []
  [left]
    type = NeumannBC
    variable = temp
    boundary = left
    value = 0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = 0.0 # -1e-8
    mass = 1
  []
[]
[Materials]
  [thermal_cond]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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'
  []
[]
(modules/solid_mechanics/test/tests/scalar_material_damage/nonlocal_scalar_damage.i)
[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]
[Mesh]
  type = GeneratedMesh
  dim = 3
  xmin = -0.5
  xmax = 0.5
  nx = 5
  ny = 5
  nz = 5
  elem_type = HEX8
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    incremental = true
    add_variables = true
    generate_output = 'stress_xx strain_xx'
  []
[]
[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
  []
  [axial_load]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.01
  []
[]
[Functions]
  [func]
    type = ParsedFunction
    expression = 'if(x>=0,0.5*t, t)'
  []
[]
[UserObjects]
  [ele_avg]
    type = RadialAverage
    prop_name = local_damage
    weights = constant
    execute_on = "INITIAL timestep_end"
    radius = 0.55
  []
[]
[Materials]
  [local_damage_index]
    type = GenericFunctionMaterial
    prop_names = local_damage_index
    prop_values = func
  []
  [local_damage]
    type = ScalarMaterialDamage
    damage_index = local_damage_index
    damage_index_name = local_damage
  []
  [damage]
    type = NonlocalDamage
    average_UO = ele_avg
    local_damage_model = local_damage
    damage_index_name = nonlocal_damage
  []
  [stress]
    type = ComputeDamageStress
    damage_model = damage
  []
  [elasticity]
    type = ComputeIsotropicElasticityTensor
    poissons_ratio = 0.2
    youngs_modulus = 10e9
  []
[]
[Postprocessors]
  [stress_xx]
    type = ElementAverageValue
    variable = stress_xx
  []
  [strain_xx]
    type = ElementAverageValue
    variable = strain_xx
  []
  [nonlocal_damage]
    type = ElementAverageMaterialProperty
    mat_prop = nonlocal_damage
  []
  [local_damage]
    type = ElementAverageMaterialProperty
    mat_prop = local_damage
  []
[]
[Executioner]
  type = Transient
  l_max_its = 50
  l_tol = 1e-8
  nl_max_its = 20
  nl_rel_tol = 1e-12
  nl_abs_tol = 1e-8
  dt = 0.2
  dtmin = 0.1
  end_time = 1
[]
[Outputs]
  csv = true
[]
(modules/combined/examples/optimization/2d_mbb.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-8
power = 2
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 150
    ny = 50
    xmin = 0
    xmax = 30
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = pull
    nodes = 0
  []
  [push]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push
    coord = '30 10 0'
  []
[]
[AuxVariables]
  [Emin]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${Emin}
  []
  [power]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${power}
  []
  [E0]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${E0}
  []
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_y
    boundary = pull
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [pull]
    type = NodalGravity
    variable = disp_y
    boundary = push
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'Emin mat_den power E0'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
    incremental = false
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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-8
  dt = 1.0
  num_steps = 70
[]
[Outputs]
  [out]
    type = Exodus
    execute_on = 'TIMESTEP_END'
  []
  print_linear_residuals = false
[]
[Postprocessors]
  [total_vol]
    type = ElementIntegralVariablePostprocessor
    variable = mat_den
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(modules/combined/test/tests/optimization/optimization_density_update/top_opt_2d.i)
vol_frac = 0.4
E0 = 1e5
Emin = 1e-4
power = 2
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 40
    ny = 20
    xmin = 0
    xmax = 20
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = pull
    nodes = 0
  []
[]
[AuxVariables]
  [sensitivity]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
    [AuxKernel]
      type = MaterialRealAux
      variable = sensitivity
      property = sensitivity
      execute_on = LINEAR
    []
  []
  [compliance]
    family = MONOMIAL
    order = CONSTANT
  []
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [pull]
    type = NodalGravity
    variable = disp_y
    boundary = pull
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # Emin + (density^penal) * (E0 - Emin)
    expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [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
    incremental = false
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.5
    weights = constant
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
    execution_order_group = -1
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
  []
[]
[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type '
  petsc_options_value = 'lu'
  nl_abs_tol = 1e-10
  start_time = 0.0
  dt = 1.0
  num_steps = 50
[]
[Outputs]
  [out]
    type = Exodus
    time_step_interval = 10
  []
[]
(modules/optimization/test/tests/simp/2d.i)
vol_frac = 0.2
[Mesh]
  [planet]
    type = ConcentricCircleMeshGenerator
    has_outer_square = false
    radii = 1
    num_sectors = 10
    rings = 2
    preserve_volumes = false
  []
  [moon]
    type = ConcentricCircleMeshGenerator
    has_outer_square = false
    radii = 0.5
    num_sectors = 8
    rings = 2
    preserve_volumes = false
  []
  [combine]
    type = CombinerGenerator
    inputs = 'planet moon'
    positions = '0 0 0 -1.5 -0.5 0'
  []
[]
[AuxVariables]
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.1
  []
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
[]
[Variables]
  [u]
  []
  [v]
  []
[]
[Kernels]
  [diff_u]
    type = Diffusion
    variable = u
  []
  [dt_u]
    type = TimeDerivative
    variable = u
  []
  [diff_v]
    type = Diffusion
    variable = v
  []
  [dt_v]
    type = TimeDerivative
    variable = v
  []
[]
[Materials]
  [thermal_cond]
    type = GenericFunctionMaterial
    prop_values = '-1.4*abs(y)-2.7*abs(x)'
    prop_names = thermal_cond
    outputs = 'exodus'
  []
  [thermal_compliance_sensitivity]
    type = GenericFunctionMaterial
    prop_values = '-3*abs(y)-1.5*abs(x)'
    prop_names = thermal_sensitivity
    outputs = 'exodus'
  []
[]
[BCs]
  [flux_u]
    type = DirichletBC
    variable = u
    boundary = outer
    value = 3.0
  []
  [flux_v]
    type = DirichletBC
    variable = v
    boundary = outer
    value = 7.0
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 0.1
    weights = linear
    prop_name = thermal_sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    execute_on = TIMESTEP_END
    force_postaux = true
  []
[]
[Executioner]
  type = Transient
  dt = 0.1
  num_steps = 15
  nl_rel_tol = 1e-04
[]
[Outputs]
  exodus = true
[]
(modules/combined/test/tests/optimization/compliance_sensitivity/2d_mmb_2material.i)
vol_frac = 0.5
power = 1
E0 = 1e-5
E1 = 0.6
E2 = 1.0
rho0 = 0.0
rho1 = 0.4
rho2 = 1.0
[GlobalParams]
  displacements = 'disp_x disp_y'
[]
[Mesh]
  [MeshGenerator]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 150
    ny = 50
    xmin = 0
    xmax = 30
    ymin = 0
    ymax = 10
  []
  [node]
    type = ExtraNodesetGenerator
    input = MeshGenerator
    new_boundary = hold
    nodes = 0
  []
  [push]
    type = ExtraNodesetGenerator
    input = node
    new_boundary = push
    coord = '30 10 0'
  []
[]
[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = CONSTANT
    # initial_condition = ${vol_frac}
  []
[]
[ICs]
  [mat_den]
    type = RandomIC
    seed = 5
    variable = mat_den
    max = '${fparse vol_frac+0.15}'
    min = '${fparse vol_frac-0.15}'
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_y]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push]
    type = NodalGravity
    variable = disp_y
    boundary = push
    gravity_value = -1
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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^${power}+B2; "
                 "if(mat_den<${rho1},E1,E2)"
    coupled_variables = 'mat_den'
    property_name = E_phys
  []
  [poissons_ratio]
    type = GenericConstantMaterial
    prop_names = poissons_ratio
    prop_values = 0.3
  []
  [stress]
    type = ComputeLinearElasticStress
  []
  [dc]
    type = ComplianceSensitivity2
    design_density = mat_den
    youngs_modulus = E_phys
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [update]
    type = DensityUpdate
    density_sensitivity = Dc
    design_density = mat_den
    volume_fraction = ${vol_frac}
    execute_on = TIMESTEP_BEGIN
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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-8
  dt = 1.0
  num_steps = 70
[]
[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'
  []
  [sensitivity]
    type = ElementIntegralMaterialProperty
    mat_prop = sensitivity
  []
[]
(modules/combined/examples/optimization/thermomechanical/structural_sub.i)
vol_frac = 0.4
power = 2.0
E0 = 1.0e-6
E1 = 1.0
rho0 = 0.0
rho1 = 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
    included_boundaries = left
  []
  [dirichlet_bc]
    type = SideSetsFromNodeSetsGenerator
    input = extra
  []
[]
[Variables]
  [disp_x]
  []
  [disp_y]
  []
[]
[AuxVariables]
  [Dc]
    family = MONOMIAL
    order = FIRST
    initial_condition = -1.0
  []
  [mat_den]
    family = MONOMIAL
    order = FIRST
    initial_condition = ${vol_frac}
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1.0e-3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1.0e-3
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    expression = "A1:=(${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
  []
  [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
  []
[]
[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]
[UserObjects]
  [rad_avg]
    type = RadialAverage
    radius = 1.2
    weights = linear
    prop_name = sensitivity
    execute_on = TIMESTEP_END
    force_preaux = true
  []
  [calc_sense]
    type = SensitivityFilter
    density_sensitivity = Dc
    design_density = mat_den
    filter_UO = rad_avg
    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]
  [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
  []
  [objective]
    type = ElementIntegralMaterialProperty
    mat_prop = strain_energy_density
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]
(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 = ${vol_frac}
  []
[]
[AuxKernels]
  [Cost]
    type = MaterialRealAux
    variable = Cost
    property = Cost_mat
  []
[]
[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    add_variables = true
    incremental = false
  []
[]
[BCs]
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = hold
    value = 0.0
  []
  [no_x_symm]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value = 0.0
  []
[]
[NodalKernels]
  [push_left]
    type = NodalGravity
    variable = disp_y
    boundary = push_left
    gravity_value = -1e-3
    mass = 1
  []
  [push_center]
    type = NodalGravity
    variable = disp_y
    boundary = push_center
    gravity_value = -1e-3
    mass = 1
  []
[]
[Materials]
  [elasticity_tensor]
    type = ComputeVariableIsotropicElasticityTensor
    youngs_modulus = E_phys
    poissons_ratio = poissons_ratio
    args = 'mat_den'
  []
  [E_phys]
    type = DerivativeParsedMaterial
    # ordered multimaterial simp
    expression = "A1:=(${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'
  []
[]