AnisotropicDiffusion

Description

The AnisotropicDiffusion kernel implements anistropic diffusion term on a domain ( $var element = document.getElementById("moose-equation-38dbd596-5a88-4280-b6a9-8e3870430b80");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ ) given in its strong form as

$(1)var element = document.getElementById("moose-equation-ff237c98-00a1-45bf-894b-84e980da736b");katex.render("\\nabla\\cdot -\\widetilde{k} \\nabla u = 0 \\in \\Omega,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-a6943133-334c-493a-aff7-13808bc776d7");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false});$ 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

$(2)var element = document.getElementById("moose-equation-7bcd7823-f5c1-401a-8f21-c831d2d0acfb");katex.render("R_i(u_h) = \\underbrace{(\\nabla \\psi_i, \\widetilde{k} \\nabla u_h)}_{\\textrm{AnisotropicDiffusion}} - \\langle\\psi_i, \\widetilde{k} \\nabla u_h \\cdot \\vec{n}\\rangle\\quad \\forall \\phi_i,", element, {displayMode:true,throwOnError:false});$ where the first term denotes the inner product over the domain volume, the latter term denotes the outward diffusion flux over the volume boundary ( $var element = document.getElementById("moose-equation-7241a6c6-307c-4e4c-8dd4-c3b6ccc2afc7");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-ae900fa9-72bd-4260-8096-d62024e9f42d");katex.render("\\psi_i", element, {displayMode:false,throwOnError:false});$ are the test functions, and $var element = document.getElementById("moose-equation-22a40fcc-e168-4718-9711-f3d4a5b6843e");katex.render("u_h \\in \\mathcal{S}^h", element, {displayMode:false,throwOnError:false});$ 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 $(3)var element = document.getElementById("moose-equation-201572ea-624f-4949-8040-c1241ef9cbe6");katex.render("\\frac{\\partial R_i(u_h)}{\\partial u_j} = (\\nabla \\phi_j, \\widetilde{k} \\nabla u_h).", element, {displayMode:true,throwOnError:false});$

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 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'
  [../]
[]

note

The anistropic diffusion coefficient $var element = document.getElementById("moose-equation-a15d3b14-5bf1-4e97-9b65-929f86d920d6");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false});$ is a three-dimensional tensor supplied through a string with nine space separated real values. The entries correspond to $var element = document.getElementById("moose-equation-0f9e18c2-f97e-496e-8aa8-f2e3ced71a2a");katex.render("xx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-152e85ca-7868-4c0e-93d1-d2fe422e4da0");katex.render("xy", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-812a21f5-3c12-4337-9ba0-1494f5c06f98");katex.render("xz", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-5d218b5d-8560-4cb0-b109-44988ba06f7a");katex.render("yx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-beba9008-8e78-4915-893d-541a9103b22e");katex.render("yy", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-3b7266fe-135b-428f-b36d-d570b5a33d0b");katex.render("yz", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-f8b5d91f-4118-46cc-8cee-6c0f61217b51");katex.render("zx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-192fcd3f-fe4d-4bab-a77a-e564fd6c8342");katex.render("zy", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-464e8bb0-dc06-45ad-92a1-f0bc39e9f1a3");katex.render("zz", element, {displayMode:false,throwOnError:false});$ , respectively. Also, this problem is 2-dimensional so the corresponding $var element = document.getElementById("moose-equation-6974539f-3cd1-4774-b9a6-b7fead76fe2c");katex.render("zx", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a910e89f-8720-45dd-8cf6-49f3961ebd2e");katex.render("zy", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-5712746f-38b9-4929-a62a-2368c109c3ee");katex.render("zz", element, {displayMode:false,throwOnError:false});$ terms are zero.

Input Parameters

variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName
Description:The name of the variable that this Kernel operates on
tensor_coeffThe Tensor to multiply the Diffusion operator by
C++ Type:libMesh::TensorValue
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
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
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
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
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
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
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
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
Description:Determines whether this object is calculated using an implicit or explicit form

Advanced Parameters

vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
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
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
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
Description:The extra tags for the matrices this Kernel should fill

Tagging Parameters

Description
Example Syntax
Input Parameters