LinearFVAnisotropicDiffusionFunctorNeumannBC

Description

LinearFVAnisotropicDiffusionFunctorNeumannBC prescribes the complete diffusive flux

var element = document.getElementById("moose-equation-95ce6e80-59ae-47f3-8adf-fd8b31d6fa5f");katex.render("g_b = \\vec{n}_b \\cdot \\mathbb{D}_b \\nabla u_b", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

for a linear finite volume variable $var element = document.getElementById("moose-equation-8cc0c4ec-3166-4405-be8e-95ffcec08156");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , where $var element = document.getElementById("moose-equation-db1164e4-b72b-4996-b8d1-b3c38aeb371f");katex.render("\\vec{n}_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the outward boundary normal and $var element = document.getElementById("moose-equation-b8758eb3-dff4-4e70-b185-e0d69ab55343");katex.render("\\mathbb{D}_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the diagonal tensor supplied through "diffusion_tensor". The flux $var element = document.getElementById("moose-equation-cd874079-cabf-4efd-9f13-5de5d6f708ee");katex.render("g_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is supplied by the "functor" parameter.

The boundary condition reconstructs the normal gradient by decomposing the tensor-weighted flux:

var element = document.getElementById("moose-equation-779e2d45-2778-4fbf-ab08-e4907215aa36");katex.render("g_b = a_n \\frac{\\partial u}{\\partial n} + \\left(\\mathbb{D}_b\\vec{n}_b-a_n\\vec{n}_b\\right)\\cdot\\nabla u_b,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-ca395f66-5e43-4181-8852-22b31ce2003e");katex.render("a_n=\\vec{n}_b\\cdot\\mathbb{D}_b\\vec{n}_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . Therefore,

var element = document.getElementById("moose-equation-c0512d37-f209-43ac-873f-b6841a2520e9");katex.render("\\frac{\\partial u}{\\partial n} = \\frac{g_b- \\left(\\mathbb{D}_b\\vec{n}_b-a_n\\vec{n}_b\\right)\\cdot\\nabla u_b}{a_n}.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

The bracketed vector is tangential to the boundary. Its contribution is used to reconstruct the normal gradient and boundary value, but it is not added to the kernel residual because the prescribed $var element = document.getElementById("moose-equation-06cb25de-45ad-4124-a3ad-f3a11d6a9101");katex.render("g_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ already contains the complete anisotropic flux. The same diffusion tensor functor should be supplied to this boundary condition and to LinearFVAnisotropicDiffusion.

By default, the boundary value is reconstructed with the normal gradient above and the tangential cell-gradient contribution. Setting "use_two_term_expansion" to false instead approximates the boundary value with the adjacent cell value and reports a zero boundary-normal gradient. The prescribed complete flux supplied to the diffusion kernel is unchanged. This one-term option avoids division by $var element = document.getElementById("moose-equation-1e17a56b-6eec-4b32-b30c-ff7ad94e0d44");katex.render("a_n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ when the boundary-normal diffusion coefficient is zero.

Input Parameters

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
diffusion_tensorFunctor describing a diagonal diffusion tensor. 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:Functor describing a diagonal diffusion tensor. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
functorThe diffusive flux functor for this boundary condition. 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 diffusive flux functor for this boundary condition. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
variableThe name of the variable that this boundary condition applies to
C++ Type:LinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this boundary condition applies to

Required Parameters

use_two_term_expansionTrueWhether to reconstruct the boundary value using the prescribed flux and cell gradient. If false, the boundary value is approximated by the adjacent cell value.
Default:True
C++ Type:bool
Controllable:No
Description:Whether to reconstruct the boundary value using the prescribed flux and cell gradient. If false, the boundary value is approximated by the adjacent cell value.

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_tagsrhsThe tag for the vectors this Kernel should fill
Default:rhs
C++ Type:MultiMooseEnum
Options:rhs, 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.
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
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).

Advanced Parameters

Input Files

(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-1d-zero-boundary-coefficient.i)
(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-2d-neumann.i)

(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-1d-zero-boundary-coefficient.i)

# A diffusion coefficient equal to x is zero only at the left boundary and remains positive on
# every interior face, allowing the constant solution to test one-term boundary reconstruction.
cells = 2

[Mesh]
  [generated]
    type = GeneratedMeshGenerator
    dim = 1
    nx = ${cells}
  []
[]

[Problem]
  linear_sys_names = u_system
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = u
    diffusion_tensor = diffusivity_tensor
  []
[]

[LinearFVBCs]
  [zero_flux]
    type = LinearFVAnisotropicDiffusionFunctorNeumannBC
    variable = u
    boundary = left
    functor = 0
    diffusion_tensor = diffusivity_tensor
    use_two_term_expansion = false
  []
  [fixed_value]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = exact_solution
  []
[]

[FunctorMaterials]
  [diffusion_tensor]
    type = GenericVectorFunctorMaterial
    prop_names = diffusivity_tensor
    prop_values = 'diffusion_x 0 0'
  []
[]

[Functions]
  [diffusion_x]
    type = ParsedFunction
    expression = x
  []
  [exact_solution]
    type = ParsedFunction
    expression = 1
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = exact_solution
  []
[]

[Executioner]
  type = Steady
  system_names = u_system
  # A direct solve isolates the boundary reconstruction from iterative solver tolerances.
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  csv = true
  execute_on = FINAL
[]

(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-2d-neumann.i)

# A rotated mesh makes the boundary-normal flux depend on both principal diffusion coefficients,
# exercising the anisotropic Neumann path rather than an axis-aligned special case.
# Thirty degrees avoids axis alignment while keeping the transformed normal easy to verify.
rotation_degrees = 30
# Two cells per direction are sufficient to exercise both boundary and internal face assembly.
cells_per_direction = 2
# Unequal coefficients and nonzero gradient components expose omitted or duplicated tensor terms.
diffusion_x = 2
diffusion_y = 3
gradient_x = 1
gradient_y = 2
rotation_radians = ${fparse rotation_degrees * pi / 180}
flux_x = ${fparse diffusion_x * gradient_x}
flux_y = ${fparse diffusion_y * gradient_y}
right_flux = ${fparse flux_x * cos(rotation_radians) + flux_y * sin(rotation_radians)}
top_flux = ${fparse -flux_x * sin(rotation_radians) + flux_y * cos(rotation_radians)}
bottom_flux = ${fparse -top_flux}

[Mesh]
  [generated]
    type = GeneratedMeshGenerator
    dim = 2
    nx = ${cells_per_direction}
    ny = ${cells_per_direction}
  []
  [rotate]
    type = TransformGenerator
    input = generated
    transform = ROTATE
    vector_value = '0 0 ${rotation_degrees}'
  []
[]

[Problem]
  linear_sys_names = u_system
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = u
    diffusion_tensor = diffusivity_tensor
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = exact_solution
  []
  [right]
    type = LinearFVAnisotropicDiffusionFunctorNeumannBC
    variable = u
    boundary = right
    functor = ${right_flux}
    diffusion_tensor = diffusivity_tensor
  []
  [top]
    type = LinearFVAnisotropicDiffusionFunctorNeumannBC
    variable = u
    boundary = top
    functor = ${top_flux}
    diffusion_tensor = diffusivity_tensor
  []
  [bottom]
    type = LinearFVAnisotropicDiffusionFunctorNeumannBC
    variable = u
    boundary = bottom
    functor = ${bottom_flux}
    diffusion_tensor = diffusivity_tensor
  []
[]

[FunctorMaterials]
  [diffusion_tensor]
    type = GenericVectorFunctorMaterial
    prop_names = diffusivity_tensor
    prop_values = '${diffusion_x} ${diffusion_y} 0'
  []
[]

[Functions]
  [exact_solution]
    type = ParsedFunction
    expression = '${gradient_x} * x + ${gradient_y} * y'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = exact_solution
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    # The tangential flux uses reconstructed gradients explicitly; twenty updates reduce that
    # reconstruction error below the exact-solution comparison tolerance.
    max_iterations = 20
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_system
  # Tight algebraic tolerances isolate the exact-solution comparison from the linear solve.
  l_tol = 1e-10
  l_abs_tol = 1e-12
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  csv = true
  execute_on = FINAL
[]

Description
Input Parameters
Input Files