LinearFVDiffusion

Description

This kernel contributes to the system matrix and the right hand side of a system which is solved for a linear finite volume variable MooseVariableLinearFVReal. The contributions can be derived using the integral of the diffusion term in the following form:

var element = document.getElementById("moose-equation-74374811-7814-459e-8976-438cdede1d4b");katex.render("-\\int\\limits_{V_C} \\nabla \\cdot D \\nabla u dV = \\sum\\limits_f \\int\\limits_{S_f} -D \\nabla u \\cdot \\vec{n} dS,", element, {displayMode:true,throwOnError:false});

where we used the divergence theorem to transform a volumetric integral over cell $var element = document.getElementById("moose-equation-882b58fb-e7af-48c6-9160-ac8644ffbbe8");katex.render("V_C", element, {displayMode:false,throwOnError:false});$ of a vector field to a sum of surface integrals over the faces of the cell. Furthermore, $var element = document.getElementById("moose-equation-0b1ca4b1-3ea6-4a6e-aa6e-2f45c7ca2222");katex.render("D", element, {displayMode:false,throwOnError:false});$ denotes a space dependent diffusion coefficient. With this, we can use the finite volume approximation for the face integrals in the following way:

var element = document.getElementById("moose-equation-3bbf6119-8376-462d-aa68-45489840ebc6");katex.render("-\\int\\limits_{S_f} D\\nabla u \\cdot \\vec{n} dS \\approx -D_f\\left(|\\vec{\\Delta}|[u_N - u_C] + \\overline{\\nabla u} \\cdot \\vec{k}\\right) |S_f|,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-018990d3-3ea3-4a25-ac67-304fc26ce9ca");katex.render("|S_f|", element, {displayMode:false,throwOnError:false});$ denotes the surface area. Vectors $var element = document.getElementById("moose-equation-38af99c7-4d19-4dec-ab14-4ef78817e37a");katex.render("\\vec{\\Delta}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-55fb1721-1856-4655-b452-75bb46fea658");katex.render("\\vec{k}", element, {displayMode:false,throwOnError:false});$ are determined to respect $var element = document.getElementById("moose-equation-592f90d0-2058-45db-b391-263fc77ab609");katex.render("\\vec{n} = \\vec{\\Delta} + \\vec{k}", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-c2be44f1-aadd-4e27-a378-1dc629cee675");katex.render("\\vec{\\Delta}", element, {displayMode:false,throwOnError:false});$ is always parallel to the line connecting the current and neighbor cell centroids. We use the over-relaxed approach for the split of the normal vector, described in Moukalled et al. (2016) and Jasak (1996) in detail. As shown above, using these two vectors, the approximate form of the normal-gradient is typically split into two terms:

$var element = document.getElementById("moose-equation-038bcafe-209a-43a6-a406-07a7cf119d9d");katex.render("|\\vec{\\Delta}|[u_N - u_C]", element, {displayMode:false,throwOnError:false});$ which describes a contribution that comes from a finite difference approximation of the gradient on orthogonal grids. Hence, it is referred to as an orthogonal contribution. For orthogonal meshes, $var element = document.getElementById("moose-equation-74d7b375-2ff0-42d6-aa97-9d0b75ef00a1");katex.render("|\\Delta|", element, {displayMode:false,throwOnError:false});$ is just $var element = document.getElementById("moose-equation-80d9c0c1-bb51-4afb-85f5-b35b7ad10a24");katex.render("\\frac{1}{|\\vec{r}_{CN}|}", element, {displayMode:false,throwOnError:false});$ where $var element = document.getElementById("moose-equation-8f6fc5b2-0ee1-46de-b138-05b921732606");katex.render("|\\vec{r}_{CN}|", element, {displayMode:false,throwOnError:false});$ is the distance between the current and neighbor cell centroids. This term contributes a ( $var element = document.getElementById("moose-equation-5447e5f4-b231-4d64-beb2-9f68210701fe");katex.render("|\\vec{\\Delta}||S_f|", element, {displayMode:false,throwOnError:false});$ ) to the diagonal and off-diagonal entries of the system matrix with different signs.
On non-orthogonal meshes, besides $var element = document.getElementById("moose-equation-35e2ea9d-002f-4b98-b692-795b80eeee47");katex.render("|\\vec{\\Delta}| = \\frac{1}{|\\vec{r}_{CN}*\\vec{n}|}", element, {displayMode:false,throwOnError:false});$ , the following correction term is needed: $var element = document.getElementById("moose-equation-3b91b614-e832-4722-a92e-0476271ef02e");katex.render("\\overline{\\nabla u} \\cdot \\vec{k}", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-d9c64322-80c6-4342-a16b-f151bcbd31f2");katex.render("\\overline{\\nabla u}", element, {displayMode:false,throwOnError:false});$ denotes the interpolated gradient at the face center computed using the cell gradients on the current and neighbor cells. This term is treated in an explicit manner meaning that it is added to the right hand side vector of the system.

For more information on the numerical representation of the diffusion term and the different techniques used for applying boundary conditions through this kernel, see Moukalled et al. (2016) and Jasak (1996).

The diffusion coefficient parameter ("diffusion_coeff") accepts anything that supports functor-based evaluations. For more information on functors in MOOSE, see Functor system.

Example input syntax

The input file below shows a pure diffusion problem on a two-dimensional domain.

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

Input Parameters

variableThe name of the variable whose linear system this object contributes to
C++ Type:LinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable whose linear system this object contributes to

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
diffusion_coeff1The diffusion coefficient. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
Default:1
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The diffusion coefficient. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
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.
use_nonorthogonal_correctionTrueIf the nonorthogonal correction should be used when computing the normal gradient.
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:If the nonorthogonal correction should be used when computing the normal gradient.

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_tagsrhsThe tag for the vectors this Kernel should fill
Default:rhs
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:rhs, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging 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_layers1The number of layers of elements to ghost.
Default:1
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

References

Hrvoje Jasak. Error analysis and estimation for the finite volume method with applications to fluid flows. PhD thesis, Imperial College London (University of London), 1996.

@phdthesis{jasak1996error,
    author = "Jasak, Hrvoje",
    title = "Error analysis and estimation for the finite volume method with applications to fluid flows.",
    year = "1996",
    school = "Imperial College London (University of London)"
}

Fadl Moukalled, L Mangani, Marwan Darwish, and others. The finite volume method in computational fluid dynamics. Volume 6. Springer, 2016.

@book{moukalled2016finite,
    author = "Moukalled, Fadl and Mangani, L and Darwish, Marwan and others",
    title = "The finite volume method in computational fluid dynamics",
    volume = "6",
    year = "2016",
    publisher = "Springer"
}

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

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right top bottom"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*(1.5-y*y)+2*x*y*(1.5-y*y)+2*(1.5-x*x)+2*x*y*(1.5-x*x)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_abs_tol = 1e-10
  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]
    type = CSV
    execute_on = FINAL
  []
[]

Description
Example input syntax
Input Parameters
References