NavierStokesStressIPHDGPrescribedTractionBC

This class weakly imposes a prescribed traction vector $var element = document.getElementById("moose-equation-72067ee0-801f-424b-bd95-97c865312931");katex.render("\\sigma\\hat{n}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . Perhaps the most common use case for this boundary condition is for a zero traction condition at outflow boundaries, as shown in the MMS channel case

[BCs<<<{"href": "../../syntax/BCs/index.html"}>>>]
  [momentum_x_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC<<<{"description": "Implements a prescribed stress boundary condition for use with a hybridized discretization of the Navier-Stokes equation", "href": "NavierStokesStressIPHDGPrescribedTractionBC.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'right'
    component<<<{"description": "number of component (0 = x, 1 = y, 2 = z)"}>>> = 0
    diffusivity<<<{"description": "The diffusivity"}>>> = 'mu'
    face_variable<<<{"description": "The face variable"}>>> = vel_bar_x
    prescribed_normal_flux<<<{"description": "The prescribed value of the flux dotted with the normal. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number."}>>> = 0
    pressure_face_variable<<<{"description": "The pressure variable that lives on element faces."}>>> = pressure_bar
    pressure_variable<<<{"description": "The pressure variable that lives on element interiors."}>>> = pressure
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = vel_x
  []
[]

and

[BCs<<<{"href": "../../syntax/BCs/index.html"}>>>]
  [momentum_y_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC<<<{"description": "Implements a prescribed stress boundary condition for use with a hybridized discretization of the Navier-Stokes equation", "href": "NavierStokesStressIPHDGPrescribedTractionBC.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'right'
    component<<<{"description": "number of component (0 = x, 1 = y, 2 = z)"}>>> = 1
    diffusivity<<<{"description": "The diffusivity"}>>> = 'mu'
    face_variable<<<{"description": "The face variable"}>>> = vel_bar_y
    prescribed_normal_flux<<<{"description": "The prescribed value of the flux dotted with the normal. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number."}>>> = 0
    pressure_face_variable<<<{"description": "The pressure variable that lives on element faces."}>>> = pressure_bar
    pressure_variable<<<{"description": "The pressure variable that lives on element interiors."}>>> = pressure
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = vel_y
  []
[]

When applying a zero traction condition at an outflow, if the flow is fully developed, then implicitly this applies a zero pressure condition.

Input Parameters

alphaThe stabilization coefficient required for discontinuous Galerkin schemes.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The stabilization coefficient required for discontinuous Galerkin schemes.
boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
componentnumber of component (0 = x, 1 = y, 2 = z)
C++ Type:unsigned int
Controllable:No
Description:number of component (0 = x, 1 = y, 2 = z)
diffusivityThe diffusivity
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The diffusivity
face_variableThe face variable
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The face variable
prescribed_normal_fluxThe prescribed value of the flux dotted with the normal. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The prescribed value of the flux dotted with the normal. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
pressure_face_variableThe pressure variable that lives on element faces.
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The pressure variable that lives on element faces.
pressure_variableThe pressure variable that lives on element interiors.
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The pressure variable that lives on element interiors.
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
search_methodnearest_node_connected_sidesChoice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).
Default:nearest_node_connected_sides
C++ Type:MooseEnum
Options:nearest_node_connected_sides, all_proximate_sides
Controllable:No
Description:Choice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).
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
skip_execution_outside_variable_domainFalseWhether to skip execution of this boundary condition when the variable it applies to is not defined on the boundary. This can facilitate setups with moving variable domains and fixed boundaries. Note that the FEProblem boundary-restricted integrity checks will also need to be turned off if using this option
Default:False
C++ Type:bool
Controllable:No
Description:Whether to skip execution of this boundary condition when the variable it applies to is not defined on the boundary. This can facilitate setups with moving variable domains and fixed boundaries. Note that the FEProblem boundary-restricted integrity checks will also need to be turned off if using this option
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/navier_stokes/test/tests/finite_element/ins/hdg/ip/paper-cases/bfs/bfs-2d-root.i)
(modules/navier_stokes/test/tests/finite_element/ins/hdg/ip/channel-flow/mms-channel.i)

(modules/navier_stokes/test/tests/finite_element/ins/hdg/ip/channel-flow/mms-channel.i)

mu = 2
rho = 2
k = FIRST
k_minus_one = CONSTANT
pressure_family = MONOMIAL

[GlobalParams]
  alpha = 6
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = -1
    ymax = 1
    nx = 2
    ny = 2
    elem_type = TRI6
  []
[]

[Variables]
  [vel_x]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [vel_y]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [pressure]
    family = ${pressure_family}
    order = ${k_minus_one}
  []
  [vel_bar_x]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [vel_bar_y]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [pressure_bar]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
[]

[HDGKernels]
  [momentum_x_advection]
    type = AdvectionIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_x_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 0
  []
  [momentum_y_advection]
    type = AdvectionIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_y_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 1
  []
  [pressure]
    type = MassContinuityIPHDGKernel
    variable = pressure
    face_variable = pressure_bar
    interior_velocity_vars = 'vel_x vel_y'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
  []
[]

[Kernels]
  [u_ffn]
    type = BodyForce
    variable = vel_x
    function = forcing_u
  []
  [v_ffn]
    type = BodyForce
    variable = vel_y
    function = forcing_v
  []
  [p_ffn]
    type = BodyForce
    variable = pressure
    function = forcing_p
  []
[]

[BCs]
  #
  # dirichlet
  #
  [momentum_x_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_x
    functor = exact_u
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_y
    functor = exact_v
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_x
    face_variable = vel_bar_x
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_u'
    diffusivity = 'mu'
    component = 0
  []
  [momentum_y_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_y
    face_variable = vel_bar_y
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_v'
    diffusivity = 'mu'
    component = 1
  []
  [mass_dirichlet]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'top bottom left'
    face_velocity_functors = 'exact_u exact_v'
    interior_velocity_vars = 'vel_x vel_y'
  []

  #
  # Neumann
  #
  [momentum_x_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_x
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_y
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 0
    diffusivity = 'mu'
    face_variable = vel_bar_x
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_x
  []
  [momentum_y_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 1
    diffusivity = 'mu'
    face_variable = vel_bar_y
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_y
  []
  [mass_neumann]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'right'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
    interior_velocity_vars = 'vel_x vel_y'
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho mu'
    prop_values = '${rho} ${mu}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = vel_x
    v = vel_y
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'sin((1/2)*y*pi)*cos((1/2)*x*pi)'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '(1/2)*pi^2*mu*sin((1/2)*y*pi)*cos((1/2)*x*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) + (1/2)*pi*rho*sin((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi)^2 - pi*rho*sin((1/2)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) - 1/4*pi*sin((1/4)*x*pi)*sin((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_v]
    type = ParsedFunction
    expression = 'sin((1/4)*x*pi)*cos((1/2)*y*pi)'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '(5/16)*pi^2*mu*sin((1/4)*x*pi)*cos((1/2)*y*pi) - pi*rho*sin((1/4)*x*pi)^2*sin((1/2)*y*pi)*cos((1/2)*y*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*x*pi)*sin((1/2)*y*pi)*cos((1/2)*y*pi) + (1/4)*pi*rho*sin((1/2)*y*pi)*cos((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi) + (3/2)*pi*cos((1/4)*x*pi)*cos((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'sin((3/2)*y*pi)*cos((1/4)*x*pi)'
  []
  [forcing_p]
    type = ParsedFunction
    expression = '(1/2)*pi*sin((1/4)*x*pi)*sin((1/2)*y*pi) + (1/2)*pi*sin((1/2)*x*pi)*sin((1/2)*y*pi)'
  []
[]

[Postprocessors]
  [L2u]
    type = ElementL2Error
    function = exact_u
    variable = vel_x
  []
  [L2v]
    type = ElementL2Error
    function = exact_v
    variable = vel_y
  []
  [L2p]
    type = ElementL2Error
    function = exact_p
    variable = pressure
  []
  [h]
    type = AverageElementSize
    execute_on = 'timestep_end'
  []
[]

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

[Outputs]
  csv = true
  [out]
    type = Exodus
    hide = 'pressure_bar vel_bar_x vel_bar_y L2u L2v L2p'
  []
[]

(modules/navier_stokes/test/tests/finite_element/ins/hdg/ip/channel-flow/mms-channel.i)

mu = 2
rho = 2
k = FIRST
k_minus_one = CONSTANT
pressure_family = MONOMIAL

[GlobalParams]
  alpha = 6
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = -1
    ymax = 1
    nx = 2
    ny = 2
    elem_type = TRI6
  []
[]

[Variables]
  [vel_x]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [vel_y]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [pressure]
    family = ${pressure_family}
    order = ${k_minus_one}
  []
  [vel_bar_x]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [vel_bar_y]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [pressure_bar]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
[]

[HDGKernels]
  [momentum_x_advection]
    type = AdvectionIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_x_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 0
  []
  [momentum_y_advection]
    type = AdvectionIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_y_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 1
  []
  [pressure]
    type = MassContinuityIPHDGKernel
    variable = pressure
    face_variable = pressure_bar
    interior_velocity_vars = 'vel_x vel_y'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
  []
[]

[Kernels]
  [u_ffn]
    type = BodyForce
    variable = vel_x
    function = forcing_u
  []
  [v_ffn]
    type = BodyForce
    variable = vel_y
    function = forcing_v
  []
  [p_ffn]
    type = BodyForce
    variable = pressure
    function = forcing_p
  []
[]

[BCs]
  #
  # dirichlet
  #
  [momentum_x_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_x
    functor = exact_u
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_y
    functor = exact_v
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_x
    face_variable = vel_bar_x
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_u'
    diffusivity = 'mu'
    component = 0
  []
  [momentum_y_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_y
    face_variable = vel_bar_y
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_v'
    diffusivity = 'mu'
    component = 1
  []
  [mass_dirichlet]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'top bottom left'
    face_velocity_functors = 'exact_u exact_v'
    interior_velocity_vars = 'vel_x vel_y'
  []

  #
  # Neumann
  #
  [momentum_x_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_x
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_y
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 0
    diffusivity = 'mu'
    face_variable = vel_bar_x
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_x
  []
  [momentum_y_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 1
    diffusivity = 'mu'
    face_variable = vel_bar_y
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_y
  []
  [mass_neumann]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'right'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
    interior_velocity_vars = 'vel_x vel_y'
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho mu'
    prop_values = '${rho} ${mu}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = vel_x
    v = vel_y
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'sin((1/2)*y*pi)*cos((1/2)*x*pi)'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '(1/2)*pi^2*mu*sin((1/2)*y*pi)*cos((1/2)*x*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) + (1/2)*pi*rho*sin((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi)^2 - pi*rho*sin((1/2)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) - 1/4*pi*sin((1/4)*x*pi)*sin((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_v]
    type = ParsedFunction
    expression = 'sin((1/4)*x*pi)*cos((1/2)*y*pi)'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '(5/16)*pi^2*mu*sin((1/4)*x*pi)*cos((1/2)*y*pi) - pi*rho*sin((1/4)*x*pi)^2*sin((1/2)*y*pi)*cos((1/2)*y*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*x*pi)*sin((1/2)*y*pi)*cos((1/2)*y*pi) + (1/4)*pi*rho*sin((1/2)*y*pi)*cos((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi) + (3/2)*pi*cos((1/4)*x*pi)*cos((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'sin((3/2)*y*pi)*cos((1/4)*x*pi)'
  []
  [forcing_p]
    type = ParsedFunction
    expression = '(1/2)*pi*sin((1/4)*x*pi)*sin((1/2)*y*pi) + (1/2)*pi*sin((1/2)*x*pi)*sin((1/2)*y*pi)'
  []
[]

[Postprocessors]
  [L2u]
    type = ElementL2Error
    function = exact_u
    variable = vel_x
  []
  [L2v]
    type = ElementL2Error
    function = exact_v
    variable = vel_y
  []
  [L2p]
    type = ElementL2Error
    function = exact_p
    variable = pressure
  []
  [h]
    type = AverageElementSize
    execute_on = 'timestep_end'
  []
[]

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

[Outputs]
  csv = true
  [out]
    type = Exodus
    hide = 'pressure_bar vel_bar_x vel_bar_y L2u L2v L2p'
  []
[]

(modules/navier_stokes/test/tests/finite_element/ins/hdg/ip/paper-cases/bfs/bfs-2d-root.i)

gamma = 1e4
degree = 2
alpha = '${fparse 10 * degree^2}'
rho = 1

[Mesh]
  [file]
    type = FileMeshGenerator
    file = coarse12.msh
  []
  second_order = true
[]

[Problem]
  type = NavierStokesProblem
  extra_tag_matrices = 'mass'
  mass_matrix = 'mass'
  set_schur_pre = A11_AND_MASS
[]

[Variables]
  [vel_x]
    family = L2_LAGRANGE
    order = SECOND
  []
  [vel_y]
    family = L2_LAGRANGE
    order = SECOND
  []
  [pressure]
    family = L2_LAGRANGE
    order = FIRST
  []
  [vel_bar_x]
    family = SIDE_HIERARCHIC
    order = SECOND
  []
  [vel_bar_y]
    family = SIDE_HIERARCHIC
    order = SECOND
  []
  [pressure_bar]
    family = SIDE_HIERARCHIC
    order = SECOND
  []
[]

[HDGKernels]
  [momentum_x_advection]
    type = AdvectionIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_x_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    diffusivity = 'mu'
    alpha = ${alpha}
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 0
  []
  [momentum_y_advection]
    type = AdvectionIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_y_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    diffusivity = 'mu'
    alpha = ${alpha}
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 1
  []
  [pressure]
    type = MassContinuityIPHDGKernel
    variable = pressure
    face_variable = pressure_bar
    interior_velocity_vars = 'vel_x vel_y'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
  []

  [u_jump]
    type = MassFluxPenaltyIPHDG
    variable = vel_x
    face_variable = vel_bar_x
    face_velocity = face_velocity
    u = vel_x
    v = vel_y
    component = 0
    gamma = ${gamma}
  []
  [v_jump]
    type = MassFluxPenaltyIPHDG
    variable = vel_y
    face_variable = vel_bar_y
    face_velocity = face_velocity
    u = vel_x
    v = vel_y
    component = 1
    gamma = ${gamma}
  []
  [pb_mass]
    type = MassMatrixHDG
    variable = pressure_bar
    matrix_tags = 'mass'
    density = '${fparse -1/gamma}'
  []
[]

[BCs]
  #
  # inlet
  #
  [momentum_x_advection_inlet]
    type = AdvectionIPHDGDirichletBC
    boundary = '1'
    face_variable = vel_bar_x
    functor = u_inlet
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_inlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = '1'
    variable = vel_x
    face_variable = vel_bar_x
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    alpha = ${alpha}
    functor = 'u_inlet'
    diffusivity = 'mu'
    component = 0
  []
  [momentum_y_diffusion_inlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = '1'
    variable = vel_y
    face_variable = vel_bar_y
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    alpha = ${alpha}
    functor = 0
    diffusivity = 'mu'
    component = 1
  []
  [mass_inlet]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = '1'
    face_velocity_functors = 'u_inlet 0'
    interior_velocity_vars = 'vel_x vel_y'
  []

  #
  # walls
  #
  [momentum_x_diffusion_walls]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = '2'
    variable = vel_x
    face_variable = vel_bar_x
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    alpha = ${alpha}
    functor = '0'
    diffusivity = 'mu'
    component = 0
  []
  [momentum_y_diffusion_walls]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = '2'
    variable = vel_y
    face_variable = vel_bar_y
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    alpha = ${alpha}
    functor = 0
    diffusivity = 'mu'
    component = 1
  []
  [mass_walls]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = '2'
    face_velocity_functors = '0 0'
    interior_velocity_vars = 'vel_x vel_y'
  []

  #
  # Neumann
  #
  [momentum_x_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = '3'
    constrain_lm = false
    face_variable = vel_bar_x
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = '3'
    constrain_lm = false
    face_variable = vel_bar_y
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = '3'
    component = 0
    diffusivity = 'mu'
    face_variable = vel_bar_x
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_x
    alpha = ${alpha}
  []
  [momentum_y_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = '3'
    component = 1
    diffusivity = 'mu'
    face_variable = vel_bar_y
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_y
    alpha = ${alpha}
  []
  [mass_neumann]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = '3'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
    interior_velocity_vars = 'vel_x vel_y'
  []

  [pb_mass]
    type = MassMatrixIntegratedBC
    variable = pressure_bar
    matrix_tags = 'mass'
    boundary = '1 2 3'
    density = '${fparse -1/gamma}'
  []

  [u_jump_walls]
    type = MassFluxPenaltyBC
    variable = vel_x
    face_variable = vel_bar_x
    u = vel_x
    v = vel_y
    component = 0
    boundary = '2'
    gamma = ${gamma}
    face_velocity = vel_walls
    dirichlet_boundary = true
  []
  [v_jump_walls]
    type = MassFluxPenaltyBC
    variable = vel_y
    face_variable = vel_bar_y
    u = vel_x
    v = vel_y
    component = 1
    boundary = '2'
    gamma = ${gamma}
    face_velocity = vel_walls
    dirichlet_boundary = true
  []
  [u_jump_inlet]
    type = MassFluxPenaltyBC
    variable = vel_x
    face_variable = vel_bar_x
    u = vel_x
    v = vel_y
    component = 0
    boundary = '1'
    gamma = ${gamma}
    face_velocity = vel_inlet
    dirichlet_boundary = true
  []
  [v_jump_inlet]
    type = MassFluxPenaltyBC
    variable = vel_y
    face_variable = vel_bar_y
    u = vel_x
    v = vel_y
    component = 1
    boundary = '1'
    gamma = ${gamma}
    face_velocity = vel_inlet
    dirichlet_boundary = true
  []
  [u_jump_outlet]
    type = MassFluxPenaltyBC
    variable = vel_x
    face_variable = vel_bar_x
    u = vel_x
    v = vel_y
    component = 0
    boundary = '3'
    gamma = ${gamma}
    face_velocity = face_velocity
    dirichlet_boundary = false
  []
  [v_jump_outlet]
    type = MassFluxPenaltyBC
    variable = vel_y
    face_variable = vel_bar_y
    u = vel_x
    v = vel_y
    component = 1
    boundary = '3'
    gamma = ${gamma}
    face_velocity = face_velocity
    dirichlet_boundary = false
  []
[]

[Materials]
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = vel_x
    v = vel_y
  []
  [mu]
    type = ADParsedMaterial
    functor_names = 'reynolds'
    functor_symbols = 'reynolds'
    property_name = 'mu'
    expression = '1 / reynolds'
  []
[]

[Functions]
  [u_inlet]
    type = ParsedFunction
    expression = '4*(2-y)*(y-1)'
  []
  [reynolds]
    type = ParsedFunction
    expression = 't'
  []
[]

[FunctorMaterials]
  [face_velocity]
    type = ADGenericVectorFunctorMaterial
    prop_names = face_velocity
    prop_values = 'vel_bar_x vel_bar_y 0'
  []
  [vel_inlet]
    type = GenericVectorFunctorMaterial
    prop_names = vel_inlet
    prop_values = 'u_inlet 0 0'
  []
  [vel_walls]
    type = GenericVectorFunctorMaterial
    prop_names = vel_walls
    prop_values = '0 0 0'
  []
[]

[Postprocessors]
  [reynolds]
    type = FunctionValuePostprocessor
    function = reynolds
  []
  [vel_mag_avg]
    type = ElementAverageValue
    variable = vel_mag
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-nl0_condensed_ksp_view'
  petsc_options_iname = '-ksp_type'
  petsc_options_value = 'preonly'
  nl_abs_tol = 1e-7
  [TimeStepper]
    type = TimeSequenceStepper
    time_sequence = '1 10 50 100 150 200 250 350 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 9200 9400 9600 9800 10000'
    use_last_t_for_end_time = true
  []
  abort_on_solve_fail = true
[]

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

[AuxVariables]
  [vel_mag]
    family = L2_LAGRANGE
    order = SECOND
  []
[]

[AuxKernels]
  [vel_mag]
    type = VectorMagnitudeAux
    variable = vel_mag
    x = vel_x
    y = vel_y
  []
[]

(modules/navier_stokes/test/tests/finite_element/ins/hdg/ip/channel-flow/mms-channel.i)

mu = 2
rho = 2
k = FIRST
k_minus_one = CONSTANT
pressure_family = MONOMIAL

[GlobalParams]
  alpha = 6
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = -1
    ymax = 1
    nx = 2
    ny = 2
    elem_type = TRI6
  []
[]

[Variables]
  [vel_x]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [vel_y]
    family = L2_HIERARCHIC
    order = ${k}
  []
  [pressure]
    family = ${pressure_family}
    order = ${k_minus_one}
  []
  [vel_bar_x]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [vel_bar_y]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
  [pressure_bar]
    family = SIDE_HIERARCHIC
    order = ${k}
  []
[]

[HDGKernels]
  [momentum_x_advection]
    type = AdvectionIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_x_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_x
    face_variable = vel_bar_x
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 0
  []
  [momentum_y_advection]
    type = AdvectionIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    velocity = 'velocity'
    coeff = ${rho}
  []
  [momentum_y_diffusion]
    type = NavierStokesStressIPHDGKernel
    variable = vel_y
    face_variable = vel_bar_y
    diffusivity = 'mu'
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    component = 1
  []
  [pressure]
    type = MassContinuityIPHDGKernel
    variable = pressure
    face_variable = pressure_bar
    interior_velocity_vars = 'vel_x vel_y'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
  []
[]

[Kernels]
  [u_ffn]
    type = BodyForce
    variable = vel_x
    function = forcing_u
  []
  [v_ffn]
    type = BodyForce
    variable = vel_y
    function = forcing_v
  []
  [p_ffn]
    type = BodyForce
    variable = pressure
    function = forcing_p
  []
[]

[BCs]
  #
  # dirichlet
  #
  [momentum_x_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_x
    functor = exact_u
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_dirichlet]
    type = AdvectionIPHDGDirichletBC
    boundary = 'top bottom left'
    face_variable = vel_bar_y
    functor = exact_v
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_x
    face_variable = vel_bar_x
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_u'
    diffusivity = 'mu'
    component = 0
  []
  [momentum_y_diffusion_dirichlet]
    type = NavierStokesStressIPHDGDirichletBC
    boundary = 'top bottom left'
    variable = vel_y
    face_variable = vel_bar_y
    pressure_variable = pressure
    pressure_face_variable = pressure_bar
    functor = 'exact_v'
    diffusivity = 'mu'
    component = 1
  []
  [mass_dirichlet]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'top bottom left'
    face_velocity_functors = 'exact_u exact_v'
    interior_velocity_vars = 'vel_x vel_y'
  []

  #
  # Neumann
  #
  [momentum_x_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_x
    variable = vel_x
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_y_advection_neumann]
    type = AdvectionIPHDGOutflowBC
    boundary = 'right'
    constrain_lm = false
    face_variable = vel_bar_y
    variable = vel_y
    velocity = velocity
    coeff = ${rho}
  []
  [momentum_x_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 0
    diffusivity = 'mu'
    face_variable = vel_bar_x
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_x
  []
  [momentum_y_diffusion_neumann]
    type = NavierStokesStressIPHDGPrescribedTractionBC
    boundary = 'right'
    component = 1
    diffusivity = 'mu'
    face_variable = vel_bar_y
    prescribed_normal_flux = 0
    pressure_face_variable = pressure_bar
    pressure_variable = pressure
    variable = vel_y
  []
  [mass_neumann]
    type = MassContinuityIPHDGBC
    face_variable = pressure_bar
    variable = pressure
    boundary = 'right'
    face_velocity_functors = 'vel_bar_x vel_bar_y'
    interior_velocity_vars = 'vel_x vel_y'
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'rho mu'
    prop_values = '${rho} ${mu}'
  []
  [vel]
    type = ADVectorFromComponentVariablesMaterial
    vector_prop_name = 'velocity'
    u = vel_x
    v = vel_y
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'sin((1/2)*y*pi)*cos((1/2)*x*pi)'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '(1/2)*pi^2*mu*sin((1/2)*y*pi)*cos((1/2)*x*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) + (1/2)*pi*rho*sin((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi)^2 - pi*rho*sin((1/2)*x*pi)*sin((1/2)*y*pi)^2*cos((1/2)*x*pi) - 1/4*pi*sin((1/4)*x*pi)*sin((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_v]
    type = ParsedFunction
    expression = 'sin((1/4)*x*pi)*cos((1/2)*y*pi)'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '(5/16)*pi^2*mu*sin((1/4)*x*pi)*cos((1/2)*y*pi) - pi*rho*sin((1/4)*x*pi)^2*sin((1/2)*y*pi)*cos((1/2)*y*pi) - 1/2*pi*rho*sin((1/4)*x*pi)*sin((1/2)*x*pi)*sin((1/2)*y*pi)*cos((1/2)*y*pi) + (1/4)*pi*rho*sin((1/2)*y*pi)*cos((1/4)*x*pi)*cos((1/2)*x*pi)*cos((1/2)*y*pi) + (3/2)*pi*cos((1/4)*x*pi)*cos((3/2)*y*pi)'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'sin((3/2)*y*pi)*cos((1/4)*x*pi)'
  []
  [forcing_p]
    type = ParsedFunction
    expression = '(1/2)*pi*sin((1/4)*x*pi)*sin((1/2)*y*pi) + (1/2)*pi*sin((1/2)*x*pi)*sin((1/2)*y*pi)'
  []
[]

[Postprocessors]
  [L2u]
    type = ElementL2Error
    function = exact_u
    variable = vel_x
  []
  [L2v]
    type = ElementL2Error
    function = exact_v
    variable = vel_y
  []
  [L2p]
    type = ElementL2Error
    function = exact_p
    variable = pressure
  []
  [h]
    type = AverageElementSize
    execute_on = 'timestep_end'
  []
[]

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

[Outputs]
  csv = true
  [out]
    type = Exodus
    hide = 'pressure_bar vel_bar_x vel_bar_y L2u L2v L2p'
  []
[]

Input Parameters
Input Files