DiffusionHDGKernel

This class implements the same physics as MatDiffusion (see note) but with a hybridized discontinuous Galerkin (HDG) discretization. For more information on HDG for a Poisson problem, please see (Cockburn et al., 2008). The weak forms implemented are a slight variation on the cited work in order to make them directly usable in downstream physics such as Navier-Stokes.

$var element = document.getElementById("moose-equation-c2401002-751a-4094-b51c-e21e17fcdc0e");katex.render("(\\vec{q}, \\vec{v})_{\\Omega_h} + (u, \\nabla \\cdot \\vec{v})_{\\Omega_h} - \\langle \\hat{u}, \\vec{n}\\cdot\\vec{v}\\rangle_{\\partial\\Omega_h} = 0\\\\ (D\\vec{q}, \\nabla\\omega)_{\\Omega_h} - \\langle D\\vec{q}\\cdot\\vec{n}, \\omega\\rangle_{\\partial\\Omega_h} + \\langle \\tau(u - \\hat{u}\\vec{n}\\cdot\\vec{n}), \\omega\\rangle_{\\partial\\Omega_h} = (f, \\omega)_{\\Omega_h}\\\\ -\\langle D\\vec{q}\\cdot\\vec{n}, \\mu\\rangle_{\\partial\\Omega_h} + \\langle \\tau(u - \\hat{u})\\vec{n}\\cdot\\vec{n}, \\mu\\rangle_{\\partial\\Omega_h} = -\\langle Dq_N, \\mu\\rangle_{\\partial\\Omega_N}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-10a6b499-8bd8-42ef-8e04-398a0d6c2250");katex.render("\\vec{q}", element, {displayMode:false,throwOnError:false});$ is the gradient field, $var element = document.getElementById("moose-equation-34918546-2d65-4d91-9a1a-b86a2e211b48");katex.render("\\vec{v}", element, {displayMode:false,throwOnError:false});$ are its associated test functions, $var element = document.getElementById("moose-equation-4ad96f3e-4523-4244-8d5c-522f95fb156e");katex.render("u", element, {displayMode:false,throwOnError:false});$ is the scalar field, $var element = document.getElementById("moose-equation-b7210997-09ed-45d2-96a9-d00bc264f742");katex.render("\\omega", element, {displayMode:false,throwOnError:false});$ are its associated test functions, $var element = document.getElementById("moose-equation-c003fbe3-7486-4f42-b61e-42c0ff01a00f");katex.render("\\hat{u}", element, {displayMode:false,throwOnError:false});$ is the trace of the scalar field (lives on the mesh skeleton), $var element = document.getElementById("moose-equation-fd3246fb-f9f3-4d1b-aa1b-5889819fdfe2");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ are its associated test functions, $var element = document.getElementById("moose-equation-d2404066-774b-42ff-963b-82dd4f8f518f");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is a stabilization parameter, $var element = document.getElementById("moose-equation-2d856c6b-8593-4f07-8d5c-995d45079436");katex.render("f", element, {displayMode:false,throwOnError:false});$ is a forcing function, $var element = document.getElementById("moose-equation-9174ef78-9dce-4ee9-a1da-c123fbeaba5e");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusivity coefficient, and $var element = document.getElementById("moose-equation-4558d6e4-c40a-4cb4-b249-065deb4af9c1");katex.render("q_N", element, {displayMode:false,throwOnError:false});$ represents a prescribed gradient on a Neumann boundary, $var element = document.getElementById("moose-equation-3fbd1438-379d-4e69-98a6-2a7a622fd3db");katex.render("\\partial\\Omega_N", element, {displayMode:false,throwOnError:false});$ . As indicated by the test functions, the first equation is the equation for the variable $var element = document.getElementById("moose-equation-e8a53b0a-b0d2-41bb-af16-f993cfd448ca");katex.render("\\vec{q}", element, {displayMode:false,throwOnError:false});$ , the second is for the variable $var element = document.getElementById("moose-equation-e94e1d9a-3bcf-4c92-844c-293afc5567f1");katex.render("u", element, {displayMode:false,throwOnError:false});$ , and the third is for the trace or Lagrange multiplier variable $var element = document.getElementById("moose-equation-2a088ff1-ab87-462c-884c-09961f5af43e");katex.render("\\hat{u}", element, {displayMode:false,throwOnError:false});$ .

$var element = document.getElementById("moose-equation-6688e40d-755d-47bc-bb5a-14cbae3ded12");katex.render("\\vec{q}", element, {displayMode:false,throwOnError:false});$ (the gradient) and $var element = document.getElementById("moose-equation-0bc04062-020e-491f-90e0-dc05ee0e7bad");katex.render("u", element, {displayMode:false,throwOnError:false});$ (the values) are referred to as the primal variables. $\hat{u} is often referred to as the dual variable. Its equation is the only one that requires the use of a linear solver. For more information on HDG solves, see the HDGKernels page.

$var element = document.getElementById("moose-equation-68ada004-e8f2-4a08-aa46-d8c46db74416");katex.render("f", element, {displayMode:false,throwOnError:false});$ the forcing function can be used for method of manufactured solutions studies, or as a source term for the diffused quantity.

note

Unlike MatDiffusion, the dependence of the diffusion coefficient on nonlinear variables is not captured in the construction of the Jacobian.

Input Parameters

diffusivityThe diffusivity
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The diffusivity
face_uThe concentration of the diffusing specie on faces
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The concentration of the diffusing specie on faces
grad_uThe gradient of the diffusing specie concentration
C++ Type:AuxVariableName
Unit:(no unit assumed)
Controllable:No
Description:The gradient of the diffusing specie concentration
uThe diffusing specie concentration
C++ Type:AuxVariableName
Unit:(no unit assumed)
Controllable:No
Description:The diffusing specie concentration
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.
source0Source for the diffusing species. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
Default:0
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Source for the diffusing species. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
tau1The stabilization coefficient required for discontinuous Galerkin schemes. This may be set to 0 for a mixed method with Raviart-Thomas.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The stabilization coefficient required for discontinuous Galerkin schemes. This may be set to 0 for a mixed method with Raviart-Thomas.
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.
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

References

Bernardo Cockburn, Bo Dong, and Johnny Guzmán. A superconvergent ldg-hybridizable galerkin method for second-order elliptic problems. Mathematics of Computation, 77(264):1887–1916, 2008.

@article{cockburn2008superconvergent,
    author = "Cockburn, Bernardo and Dong, Bo and Guzm{\'a}n, Johnny",
    title = "A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems",
    journal = "Mathematics of Computation",
    volume = "77",
    number = "264",
    pages = "1887--1916",
    year = "2008"
}

Input Parameters
References