# AnisotropicDiffusion

## Description

The AnisotropicDiffusion kernel implements anistropic diffusion term on a domain () given in its strong form as

(1) where is the anisotropic diffusion coefficient. Diffusion is anistropic if the diffusion rate varies with direction. The corresponding weak form, using inner-product notation, is given by

(2) where the first term denotes the inner product over the domain volume, the latter term denotes the outward diffusion flux over the volume boundary (), are the test functions, and 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 (3)

## 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 AnistropicDiffusion syntax in a steady-state anistropic diffusion problem can be found below:

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

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

The anistropic diffusion coefficient is a three-dimensional tensor supplied through a string with nine space separated real values. The entries correspond to , , , , , , , , and , respectively. Also, this problem is 2-dimensional so the corresponding , , and terms are zero.

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

Options:

Description:The name of the variable that this Kernel operates on

• tensor_coeffThe Tensor to multiply the Diffusion operator by

C++ Type:libMesh::TensorValue

Options:

Description:The Tensor to multiply the Diffusion operator by

### Required Parameters

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Options:

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.

• 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

Options:

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.)

• 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

Options:

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.

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

C++ Type:std::vector

Options:

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Options:

Description:The seed for the master random number generator

• 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

Options:

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.)

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Options:

Description:Determines whether this object is calculated using an implicit or explicit form

• vector_tagsnontimeThe tag for the vectors this Kernel should fill

Default:nontime

C++ Type:MultiMooseEnum

Options:nontime time

Description:The tag for the vectors this Kernel should fill

• extra_vector_tagsThe extra tags for the vectors this Kernel should fill

C++ Type:std::vector

Options:

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

Description:The tag for the matrices this Kernel should fill

• extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

C++ Type:std::vector

Options:

Description:The extra tags for the matrices this Kernel should fill