FVDiffusion

Computes residual for diffusion operator for finite volume method.

The steady-state diffusion equation on a domain $var element = document.getElementById("moose-equation-78e92a34-e65a-4754-a769-9948e6de265f");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ is defined as

var element = document.getElementById("moose-equation-8e7553c1-95b6-4aed-8ba6-f70002bc67d9");katex.render("-\\nabla \\cdot D \\nabla u = 0 \\in \\Omega.", element, {displayMode:true,throwOnError:false});

with $var element = document.getElementById("moose-equation-67bb24c9-440d-411c-8c13-26e62c5d3c8e");katex.render("D", element, {displayMode:false,throwOnError:false});$ the diffusion coefficient or diffusivity. $var element = document.getElementById("moose-equation-dd3b0034-836c-42a1-b437-24a5ae5ed6c2");katex.render("D", element, {displayMode:false,throwOnError:false});$ has to be supplied as material property to this kernel.

The diffusion term is integrated using the divergence theorem, turning it from a volumetric second order derivative term into a first order derivative integrated over a surface.

var element = document.getElementById("moose-equation-5c43f5e8-2603-490f-99df-7fca84b8a2c5");katex.render("\\int_{element} -\\nabla \\cdot D \\nabla u = \\sum_{elem faces f} -D_f \\nabla u_f \\cdot \\vec{n_f} area_f", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-2ce7c45e-8bc9-44e8-ba6b-cc6dedb7d05f");katex.render("\\vec{n_f}", element, {displayMode:false,throwOnError:false});$ is the surface normal on each side of the element considered.

The diffusion coefficient can be interpolated to the surface using two approaches:

Simple arithmetic average: $var element = document.getElementById("moose-equation-e42f4b59-5c21-4ce4-956b-afaa4033ac76");katex.render("D_f = w_1 D_1 + (1-w_1) D_2", element, {displayMode:false,throwOnError:false});$ (with $var element = document.getElementById("moose-equation-a0ddae64-dc68-4ace-8e69-38c2e9663615");katex.render("D_1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-68c6a292-acd2-45fa-8ca0-f3b510eef641");katex.render("D_2", element, {displayMode:false,throwOnError:false});$ being the diffusion coefficient in the neighboring cells respectively)
Simple harmonic average: $var element = document.getElementById("moose-equation-3cba7508-1d15-4bfc-b42b-2cd87ede3f84");katex.render("D_f = \\frac{1}{\\frac{w_1}{D_1} + \\frac{1 - w_1}{D_2}}", element, {displayMode:false,throwOnError:false});$ , which yields better results if the diffusion coefficients are positive and discontinuous. This is due to the fact that this scheme preserves flux continuity in the face-normal direction on orthogonal grids.

The interpolation method can be set using the "coeff_interp_method" parameter, and is defaulted to harmonic due to its superior accuracy for discontinuous diffusion coefficients. Simple tests cases with discontinuous diffusion coefficients (see below) indicate that using harmonic interpolation yields a second-order accurate scheme for orthogonal and 1D meshes and close to second-order accurate scheme for slightly non-orthogonal meshes. At the same time, using a simple arithmetic average for the interpolation of discontinuous diffusion coefficients yields a first order scheme.

note

This kernel leverages the automatic differentiation system, so the Jacobian is computed at the same time as the residual and need not be defined separately.

Example input syntax

This example shows a simple 1D diffusion problem with two variables defined on two subdomains. Because of the limits of the legacy material system, the two material properties have to have different names, otherwise it is not clear what the boundary value of the diffusion coefficient should be.

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 20
    xmax = 2
  []
  [subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  []
  [left_right]
    input = subdomain1
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = '0'
    paired_block = '1'
    new_boundary = 'left_right'
  []
  [right_left]
    input = left_right
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = '1'
    paired_block = '0'
    new_boundary = 'right_left'
  []
[]

[Variables]
  [left]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    block = 0
  []
  [right]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    block = 1
  []
[]

[FVKernels]
  [left]
    type = FVDiffusion
    variable = left
    coeff = coeff_left
    block = 0
    coeff_interp_method = average
  []
  [right]
    type = FVDiffusion
    variable = right
    coeff = coeff_right
    block = 1
    coeff_interp_method = average
  []
[]

[FVBCs]
  [left]
    type = FVDirichletBC
    variable = left
    boundary = left
    value = 0
  []
  [left_right]
    type = FVDirichletBC
    variable = left
    boundary = left_right
    value = 1
  []
  [right_left]
    type = FVDirichletBC
    variable = right
    boundary = right_left
    value = 0
  []
  [right]
    type = FVDirichletBC
    variable = right
    boundary = right
    value = 1
  []
[]

[Materials]
  [left]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff_left'
    prop_values = '1'
    block = 0
  []
  [right]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff_right'
    prop_values = '1'
    block = 1
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  line_search = 'none'
[]

[Outputs]
  exodus = true
[]

Input Parameters

coeffdiffusion coefficient. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:diffusion coefficient. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
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

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Unit:(no unit assumed)
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
coeff_interp_methodharmonicSwitch that can select face interpolation method for diffusion coefficients.
Default:harmonic
C++ Type:MooseEnum
Unit:(no unit assumed)
Options:average, harmonic
Controllable:No
Description:Switch that can select face interpolation method for diffusion coefficients.
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
Unit:(no unit assumed)
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.

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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
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
Unit:(no unit assumed)
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

boundaries_to_avoidThe set of sidesets to not execute this FVFluxKernel on. This takes precedence over force_boundary_execution to restrict to less external boundaries. By default flux kernels are executed on all internal boundaries and Dirichlet boundary conditions.
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The set of sidesets to not execute this FVFluxKernel on. This takes precedence over force_boundary_execution to restrict to less external boundaries. By default flux kernels are executed on all internal boundaries and Dirichlet boundary conditions.
boundaries_to_forceThe set of sidesets to force execution of this FVFluxKernel on. Setting force_boundary_execution to true is equivalent to listing all external mesh boundaries in this parameter.
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The set of sidesets to force execution of this FVFluxKernel on. Setting force_boundary_execution to true is equivalent to listing all external mesh boundaries in this parameter.
force_boundary_executionFalseWhether to force execution of this object on all external boundaries.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Whether to force execution of this object on all external boundaries.

Boundary Execution Modification Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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

ghost_layers2The number of layers of elements to ghost.
Default:2
C++ Type:unsigned short
Unit:(no unit assumed)
Controllable:No
Description:The number of layers of elements to ghost.
use_point_neighborsFalseWhether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

Parallel Ghosting Parameters

(moose/test/tests/fvkernels/block-restriction/1d.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 20
    xmax = 2
  []
  [subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  []
  [left_right]
    input = subdomain1
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = '0'
    paired_block = '1'
    new_boundary = 'left_right'
  []
  [right_left]
    input = left_right
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = '1'
    paired_block = '0'
    new_boundary = 'right_left'
  []
[]

[Variables]
  [left]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    block = 0
  []
  [right]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    block = 1
  []
[]

[FVKernels]
  [left]
    type = FVDiffusion
    variable = left
    coeff = coeff_left
    block = 0
    coeff_interp_method = average
  []
  [right]
    type = FVDiffusion
    variable = right
    coeff = coeff_right
    block = 1
    coeff_interp_method = average
  []
[]

[FVBCs]
  [left]
    type = FVDirichletBC
    variable = left
    boundary = left
    value = 0
  []
  [left_right]
    type = FVDirichletBC
    variable = left
    boundary = left_right
    value = 1
  []
  [right_left]
    type = FVDirichletBC
    variable = right
    boundary = right_left
    value = 0
  []
  [right]
    type = FVDirichletBC
    variable = right
    boundary = right
    value = 1
  []
[]

[Materials]
  [left]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff_left'
    prop_values = '1'
    block = 0
  []
  [right]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff_right'
    prop_values = '1'
    block = 1
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  line_search = 'none'
[]

[Outputs]
  exodus = true
[]

Example input syntax
Input Parameters