LinearFVAnisotropicDiffusion

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. The difference between this kernel and LinearFVDiffusion is that this kernel requires a vector of diffusion coefficients, where every entry describes the diffusion coefficient for a principal direction. This is equivalent to supplying a diagonal tensor in the fully anisotropic diffusion case.

The implementation in this kernel is based on the derivation in Liu et al. (2015). The contributions of the system matrix and right hand side can be derived using the divergence theorem on a volumetric integral over cell $var element = document.getElementById("moose-equation-e8b2fe9b-a838-45fb-bcaf-da4fdb57ede6");katex.render("C", element, {displayMode:false,throwOnError:false});$ :

var element = document.getElementById("moose-equation-3f071a72-ae00-40af-88c3-61602053c555");katex.render("\\int\\limits_{V_C} \\nabla \\cdot \\mathbb{D} \\nabla u dV = \\sum\\limits_f \\int\\limits_{S_f} \\mathbb{D} \\nabla u \\cdot \\vec{n} dS,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-4e27d25c-ec84-48ae-b493-716f662c2a03");katex.render("\\mathbb{D}", element, {displayMode:false,throwOnError:false});$ denotes a space dependent diagonal diffusion tensor, while the right hand side describes the sum of the surface integrals on each side of cell $var element = document.getElementById("moose-equation-2dfef57d-0329-4f04-adda-763f5c9ccb2a");katex.render("C", element, {displayMode:false,throwOnError:false});$ . Following Liu et al. (2015) and using the assumption that the tensor is diagonal, we can manipulate this expression to arrive to the following form:

var element = document.getElementById("moose-equation-f232b26b-f39b-4b3e-bf9d-b03ab96627b5");katex.render("\\sum\\limits_f \\int\\limits_{S_f} \\mathbb{D} \\vec{n} \\cdot \\nabla u dS,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-2e58dceb-f424-4bd0-bc79-093908640839");katex.render("mathbb{D} \\vec{n}", element, {displayMode:false,throwOnError:false});$ can be split into two contributions:

var element = document.getElementById("moose-equation-67346188-39d1-4758-aa17-ed0177671da7");katex.render("\\mathbb{D} \\vec{n} = (\\mathbb{D}\\vec{n}\\cdot \\vec{n}) \\vec{n} + (\\mathbb{D}\\vec{n} - \\mathbb{D}\\vec{n}\\cdot \\vec{n} \\vec{n}).", element, {displayMode:true,throwOnError:false});

Plugging this expression back to the surface integral, we get the following:

var element = document.getElementById("moose-equation-e7285518-5716-4073-8a5a-99eef05dbab7");katex.render("\\sum\\limits_f \\int\\limits_{S_f} (\\mathbb{D}\\vec{n}\\cdot \\vec{n}) \\vec{n} \\cdot \\nabla u + (\\mathbb{D}\\vec{n} - \\mathbb{D}\\vec{n}\\cdot \\vec{n} \\vec{n}) \\cdot \\nabla u dS,", element, {displayMode:true,throwOnError:false});

where we can treat the normal projection ( $var element = document.getElementById("moose-equation-a7c53453-7255-48b7-8f3e-7f25c9b2e1d8");katex.render("\\int\\limits_{S_f} (\\mathbb{D}\\vec{n}\\cdot \\vec{n}) \\vec{n} \\cdot \\nabla u dS", element, {displayMode:false,throwOnError:false});$ ) the same way as described in LinearFVDiffusion with $var element = document.getElementById("moose-equation-9015d67e-f1e4-4f9d-b60b-9f044c5a779c");katex.render("\\mathbb{D}\\vec{n}\\cdot \\vec{n}", element, {displayMode:false,throwOnError:false});$ replacing the diffusion coefficient. The second term ( $var element = document.getElementById("moose-equation-2a3e53b2-7b57-4674-a46b-8f82a0d0480f");katex.render("\\int\\limits_{S_f} (\\mathbb{D}\\vec{n} - \\mathbb{D}\\vec{n}\\cdot \\vec{n} \\vec{n}) \\cdot \\nabla u dS dS", element, {displayMode:false,throwOnError:false});$ ) can be treated explicitly, similarly to the nonorthogonal correction in LinearFVDiffusion.

Input Parameters

diffusion_tensorFunctor describing a diagonal diffusion tensor. 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:Functor describing a diagonal diffusion tensor. 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 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
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.
use_nonorthogonal_correction_on_boundaryFalseIf the nonorthogonal correction should be used when computing the normal gradient.
Default:False
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

Xiaogang Liu, Pingjian Ming, Wenping Zhang, Lirong Fu, and Lilong Jing. Finite-volume methods for anisotropic diffusion problems on skewed meshes. Numerical Heat Transfer, Part B: Fundamentals, 68(3):239–256, 2015.

@article{liu2015finite,
    author = "Liu, Xiaogang and Ming, Pingjian and Zhang, Wenping and Fu, Lirong and Jing, Lilong",
    title = "Finite-volume methods for anisotropic diffusion problems on skewed meshes",
    journal = "Numerical Heat Transfer, Part B: Fundamentals",
    volume = "68",
    number = "3",
    pages = "239--256",
    year = "2015",
    publisher = "Taylor \\& Francis"
}

Description
Input Parameters
References