AnisotropicDiffusion

Description

The AnisotropicDiffusion kernel implements anisotropic diffusion term on a domain ( $var element = document.getElementById("moose-equation-ffca26bc-eff4-4560-9efd-39d48d78a1d0");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ ) given in its strong form as

$var element = document.getElementById("moose-equation-7f20d3c1-9896-4d13-8554-5fcb906d5e01");katex.render("\\nabla\\cdot -\\widetilde{k} \\nabla u = 0 \\in \\Omega,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-911a89a6-4cbb-4e98-9aa6-361f8bb1fbe0");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false});$ is the anisotropic diffusion coefficient. Diffusion is anisotropic if the diffusion rate varies with direction. The corresponding weak form, using inner-product notation, is given by

$var element = document.getElementById("moose-equation-b4b822af-d153-43be-9f8a-35ba0a82548c");katex.render("R_i(u_h) = \\underbrace{(\\nabla \\psi_i, \\widetilde{k} \\nabla u_h)}_{\\textrm{AnisotropicDiffusion}} - \\langle\\psi_i, \\widetilde{k} \\nabla u_h \\cdot \\vec{n}\\rangle\\quad \\forall \\phi_i,", element, {displayMode:true,throwOnError:false});$ where the first term denotes the inner product over the domain volume, the latter term denotes the outward diffusion flux over the volume boundary ( $var element = document.getElementById("moose-equation-9b96701f-b826-41f7-a54a-4cc841f5158e");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-4bc6c897-6d58-46e9-8d43-1e98282eaee5");katex.render("\\psi_i", element, {displayMode:false,throwOnError:false});$ are the test functions, and $var element = document.getElementById("moose-equation-837c6feb-f2ca-4e85-9c76-8f114882248c");katex.render("u_h \\in \\mathcal{S}^h", element, {displayMode:false,throwOnError:false});$ is the finite element solution of the weak formulation. The AnisotropicDiffusion kernel implements the first (volume) term.

For a constant diffusion coefficient, the Jacobian is given by $var element = document.getElementById("moose-equation-61752c6c-ed88-4838-bc36-bf10b50b0339");katex.render("\\frac{\\partial R_i(u_h)}{\\partial u_j} = (\\nabla \\phi_j, \\widetilde{k} \\nabla u_h).", element, {displayMode:true,throwOnError:false});$

Example Syntax

The AnisotropicDiffusion kernel may be used in a variety of physical models, including steady-state and time-dependent diffusion, advection-diffusion-reaction, etc. A kernel block demonstrating the AnisotropicDiffusion syntax in a steady-state anisotropic diffusion problem can be found below:

[Kernels]
  [./diff]
    type = AnisotropicDiffusion
    variable = u
    tensor_coeff = '2 0 0
                    0 4 0
        0 0 0'
  [../]
[]

note

The anisotropic diffusion coefficient $var element = document.getElementById("moose-equation-2e37b520-b1b1-4b73-8675-3e36afbe772f");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false});$ is a three-dimensional tensor supplied through a string with nine space separated real values. The entries correspond to $var element = document.getElementById("moose-equation-c50598a3-0901-466a-b0c5-bc0759629d10");katex.render("xx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-11998e1e-5db7-4d32-a36c-802697a0830a");katex.render("xy", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-f96581b3-fb1b-45fd-88e4-9af2428c5257");katex.render("xz", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-bb2fac2e-926b-4f11-97f9-76bcde611b61");katex.render("yx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-171be2ca-ba55-446f-89e2-8b8a732d91c3");katex.render("yy", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-7956a99c-992f-42ab-b123-cb8155f9dc32");katex.render("yz", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-d103bb38-59d6-4fd2-8f56-8f95e4ffa199");katex.render("zx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-6cdfdc84-1d7e-4fc1-a02a-8014fd98923f");katex.render("zy", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-18735edd-cce3-4584-a3c3-2af41d31c232");katex.render("zz", element, {displayMode:false,throwOnError:false});$ , respectively. Also, this problem is 2-dimensional so the corresponding $var element = document.getElementById("moose-equation-363a0634-ead2-44fc-8526-410f789135c1");katex.render("zx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-fec085fe-0ae1-4abd-ba1a-0a14d62c54f5");katex.render("zy", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-8f36ab61-d480-40c0-abc4-9262cfa5abb5");katex.render("zz", element, {displayMode:false,throwOnError:false});$ terms are zero.

Input Parameters

tensor_coeffThe Tensor to multiply the Diffusion operator by
C++ Type:libMesh::TensorValue<double>
Unit:(no unit assumed)
Controllable:No
Description:The Tensor to multiply the Diffusion operator by
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
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
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

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.
diag_save_inThe name of auxiliary variables to save this Kernel'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 Kernel'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
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
save_inThe name of auxiliary variables to save this Kernel'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 Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
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

(moose/test/tests/kernels/anisotropic_diffusion/aniso_diffusion.i)

[Mesh]
  file = mixed_block.e
  uniform_refine=3
[]

[Functions]
  [./top_bc]
    type = ParsedFunction
    expression = 'x'
  [../]
[]

[Variables]
  active = 'u'

  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = AnisotropicDiffusion
    variable = u
    tensor_coeff = '2 0 0
                    0 4 0
        0 0 0'
  [../]
[]

[BCs]
  active = 'lower_left top'

  [./lower_left]
    type = DirichletBC
    variable = u
    boundary = '1 4'
    value = 1
  [../]

  [./top]
    type = FunctionNeumannBC
    variable = u
    boundary = 3
    function = top_bc
  [../]

  [./right]
    type = NeumannBC
    variable = u
    boundary = 2
  [../]
[]

[Executioner]
  type = Steady

  solve_type = 'PJFNK'
[]

[Outputs]
  exodus = true
[]

Description
Example Syntax
Input Parameters