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 CC:

∫VC∇⋅D∇udV=∑f∫SfD∇u⋅n⃗dS,\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,

where D\mathbb{D} denotes a space dependent diagonal diffusion tensor, while the right hand side describes the sum of the surface integrals on each side of cell CC. 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:

∑f∫SfDn⃗⋅∇udS,\sum\limits_f \int\limits_{S_f} \mathbb{D} \vec{n} \cdot \nabla u dS,

where mathbbDn⃗mathbb{D} \vec{n} can be split into two contributions:

Dn⃗=(Dn⃗⋅n⃗)n⃗+(Dn⃗−Dn⃗⋅n⃗n⃗).\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}).

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

∑f∫Sf(Dn⃗⋅n⃗)n⃗⋅∇u+(Dn⃗−Dn⃗⋅n⃗n⃗)⋅∇udS,\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,

where we can treat the normal projection (∫Sf(Dn⃗⋅n⃗)n⃗⋅∇udS\int\limits_{S_f} (\mathbb{D}\vec{n}\cdot \vec{n}) \vec{n} \cdot \nabla u dS) the same way as described in LinearFVDiffusion with Dn⃗⋅n⃗\mathbb{D}\vec{n}\cdot \vec{n} replacing the diffusion coefficient. The second term (∫Sf(Dn⃗−Dn⃗⋅n⃗n⃗)⋅∇udSdS\int\limits_{S_f} (\mathbb{D}\vec{n} - \mathbb{D}\vec{n}\cdot \vec{n} \vec{n}) \cdot \nabla u dS dS) 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 postprocessor 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 postprocessor 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>

    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

    Controllable:No

    Description:Whether to force execution of this object on all external boundaries.

  • matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether this object is only doing assembly to matrices (no vectors)

  • use_nonorthogonal_correctionTrueIf the nonorthogonal correction should be used when computing the normal gradient.

    Default:True

    C++ Type:bool

    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

    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>

    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>

    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>

    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

    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

    Options:rhs, time

    Controllable:No

    Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    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

    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

    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

    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

    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

    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

    Controllable:No

    Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

Parallel Ghosting Parameters

Input Files

References

  1. 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.[BibTeX]