FVAnisotropicDiffusion

Computes residual for anisotropic diffusion operator for finite volume method.

An anisotropic diffusion term that is discretized using the finite-volume method:

where is the diagonal tensor diffusion coefficient and is the diffusing variable. The discretized form of the equation above over en element is the following:

where denotes the surface vector of face of the element. Furthermore, the face gradient, (\nabla \phi)_f, is determined using a central difference scheme combined with non-orthogonal correction. Lastly, components of the diffusion (diagonal) tensor can be either interpolated to the face using a geometric arithmetic average:

or a harmonic average:

where is the interpolation weight, and subscripts and represent element and neighbor values.

[Mesh<<<{"href": "../../syntax/Mesh/index.html"}>>>]
  [cmg]
    type = CartesianMeshGenerator<<<{"description": "This CartesianMeshGenerator creates a non-uniform Cartesian mesh.", "href": "../meshgenerators/CartesianMeshGenerator.html"}>>>
    dim<<<{"description": "The dimension of the mesh to be generated"}>>> = 2
    dx<<<{"description": "Intervals in the X direction"}>>> = '10 10'
    ix<<<{"description": "Number of grids in all intervals in the X direction (default to all one)"}>>> = '2 2'
    dy<<<{"description": "Intervals in the Y direction (required when dim>1 otherwise ignored)"}>>> = '20'
    iy<<<{"description": "Number of grids in all intervals in the Y direction (default to all one)"}>>> = '4'
    subdomain_id<<<{"description": "Block IDs (default to all zero)"}>>> = '1 2'
  []
[]

[Variables<<<{"href": "../../syntax/Variables/index.html"}>>>]
  [v]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    fv = true
  []

  [u]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  []
[]

[Kernels<<<{"href": "../../syntax/Kernels/index.html"}>>>]
  [fem_diff1]
    type = AnisotropicDiffusion<<<{"description": "Anisotropic diffusion kernel $\\nabla \\cdot -\\widetilde{k} \\nabla u$ with weak form given by $(\\nabla \\psi_i, \\widetilde{k} \\nabla u)$.", "href": "../kernels/AnisotropicDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = u
    tensor_coeff<<<{"description": "The Tensor to multiply the Diffusion operator by"}>>> = '1 0 0
                    0 10 0
                    0 0 0'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []

  [fem_diff2]
    type = AnisotropicDiffusion<<<{"description": "Anisotropic diffusion kernel $\\nabla \\cdot -\\widetilde{k} \\nabla u$ with weak form given by $(\\nabla \\psi_i, \\widetilde{k} \\nabla u)$.", "href": "../kernels/AnisotropicDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = u
    tensor_coeff<<<{"description": "The Tensor to multiply the Diffusion operator by"}>>> = '10 0 0
                    0 10 0
                    0 0 0'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
[]

[BCs<<<{"href": "../../syntax/BCs/index.html"}>>>]
  [fem_left_bottom]
    type = NeumannBC<<<{"description": "Imposes the integrated boundary condition $\\frac{\\partial u}{\\partial n}=h$, where $h$ is a constant, controllable value.", "href": "../bcs/NeumannBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = u
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'left bottom'
    value<<<{"description": "For a Laplacian problem, the value of the gradient dotted with the normals on the boundary."}>>> = 1
  []
  [fem_top_right]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = u
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'right top'
    value<<<{"description": "Value of the BC"}>>> = 0
  []
[]

[FVKernels<<<{"href": "../../syntax/FVKernels/index.html"}>>>]
  [diff]
    type = FVAnisotropicDiffusion<<<{"description": "Computes residual for anisotropic diffusion operator for finite volume method.", "href": "FVAnisotropicDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = v
    coeff<<<{"description": "The diagonal coefficients of a diffusion tensor. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number."}>>> = coeff
  []
[]

[FVBCs<<<{"href": "../../syntax/FVBCs/index.html"}>>>]
  [left_bottom]
    type = FVNeumannBC<<<{"description": "Neumann boundary condition for finite volume method.", "href": "../fvbcs/FVNeumannBC.html"}>>>
    variable<<<{"description": "The name of the variable that this boundary condition applies to"}>>> = v
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'left bottom'
    value<<<{"description": "The value of the flux crossing the boundary."}>>> = 1
  []
  [top_right]
    type = FVDirichletBC<<<{"description": "Defines a Dirichlet boundary condition for finite volume method.", "href": "../fvbcs/FVDirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this boundary condition applies to"}>>> = v
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'right top'
    value<<<{"description": "value to enforce at the boundary face"}>>> = 0
  []
[]

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [diff1]
    type = ADGenericVectorFunctorMaterial<<<{"description": "FunctorMaterial object for declaring vector properties that are populated by evaluation of functor (constants, functions, variables, matprops) object.", "href": "../functormaterials/GenericVectorFunctorMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = 'coeff'
    prop_values<<<{"description": "The corresponding names of the functors that are going to provide the values for the vector material properties"}>>> = '1 10 1'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []

  [diff2]
    type = ADGenericVectorFunctorMaterial<<<{"description": "FunctorMaterial object for declaring vector properties that are populated by evaluation of functor (constants, functions, variables, matprops) object.", "href": "../functormaterials/GenericVectorFunctorMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = 'coeff'
    prop_values<<<{"description": "The corresponding names of the functors that are going to provide the values for the vector material properties"}>>> = '10 10 1'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
[]

[Executioner<<<{"href": "../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs<<<{"href": "../../syntax/Outputs/index.html"}>>>]
  exodus<<<{"description": "Output the results using the default settings for Exodus output."}>>> = true
[]