DiffusionCG

Discretizes a diffusion equation with the continuous Galerkin finite element method

See the DiffusionPhysicsBase documentation for the diffusion equation solved.

Continuous Galerkin Discretization

The volumetric discretization of the time derivative uses the TimeDerivative kernel. The diffusion term $var element = document.getElementById("moose-equation-5a83eafd-732b-48d2-b75f-1dbc05fa59bf");katex.render("\\nabla \\cdot D \\nabla u(\\vec{x})", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is integrated by parts and represented using:

a Diffusion kernel if no diffusion coefficient is passed
a MatDiffusion kernel if a diffusion coefficient is passed, usually as a material property

The source term $var element = document.getElementById("moose-equation-4cd98c8a-c1b6-4b9a-972b-555632893c95");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is added using:

a BodyForce if the source is a constant, a Postprocessor or a Function
a MatCoupledForce if the force is specified using a material property
a CoupledForce / ADCoupledForce if the source is another type of functor

The Dirichlet boundary conditions are created using:

a DirichletBC if the boundary value is set to a number
a FunctionDirichletBC if set to a Function
a PostprocessorDirichletBC if set to a Postprocessor
a FunctorDirichletBC for any other kind of functor for the boundary value

The Neumann boundary conditions are created using:

a NeumannBC if the flux is set to a number
a FunctionNeumannBC if set to a Function
a PostprocessorNeumannBC if set to a Postprocessor
a FunctorNeumannBC for any other kind of functor for the flux value

note

We could use a Functor object to cover every need, but the specialized objects are a few percent faster, depending on the case.

note

The user may switch between using / not using automatic differentiation for the kernel and boundary conditions using the "use_automatic_differentiation". This parameter is only obeyed if the AD/non-AD object exists for the requested diffusivity / source / boundary value functor etc.

Input Parameters

blockBlocks (subdomains) that this Physics is active on.
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:Blocks (subdomains) that this Physics is active on.
diffusivity_functorFunctor specifying the diffusivity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Controllable:No
Description:Functor specifying the diffusivity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
diffusivity_matpropMaterial property defining the diffusion coefficient
C++ Type:MaterialPropertyName
Controllable:No
Description:Material property defining the diffusion coefficient
preconditioningdefaultWhich preconditioning to use for this Physics
Default:default
C++ Type:MooseEnum
Options:default, none
Controllable:No
Description:Which preconditioning to use for this Physics
source_coef1Coefficient multiplying the source
Default:1
C++ Type:double
Controllable:No
Description:Coefficient multiplying the source
source_functorSource term in the diffusion problem. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Controllable:No
Description:Source term in the diffusion problem. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
transientsame_as_problemWhether the physics is to be solved as a transient
Default:same_as_problem
C++ Type:MooseEnum
Options:true, false, same_as_problem
Controllable:No
Description:Whether the physics is to be solved as a transient
use_automatic_differentiationTrueWhether to use automatic differentiation for all the terms in the equation
Default:True
C++ Type:bool
Controllable:No
Description:Whether to use automatic differentiation for all the terms in the equation
variable_nameuVariable name for the equation
Default:u
C++ Type:VariableName
Controllable:No
Description:Variable name for the equation
variable_orderFIRSTOrder of the FE shape function to use for this variable (additional orders not listed here are allowed, depending on the family).
Default:FIRST
C++ Type:MooseEnum
Options:CONSTANT, FIRST, SECOND, THIRD, FOURTH, FIFTH, SIXTH, SEVENTH, EIGHTH, NINTH, TENTH, ELEVENTH, TWELFTH, THIRTEENTH, FOURTEENTH, FIFTEENTH, SIXTEENTH, SEVENTEENTH, EIGHTTEENTH, NINETEENTH, TWENTIETH, TWENTYFIRST, TWENTYSECOND, TWENTYTHIRD, TWENTYFOURTH, TWENTYFIFTH, TWENTYSIXTH, TWENTYSEVENTH, TWENTYEIGHTH, TWENTYNINTH, THIRTIETH, THIRTYFIRST, THIRTYSECOND, THIRTYTHIRD, THIRTYFOURTH, THIRTYFIFTH, THIRTYSIXTH, THIRTYSEVENTH, THIRTYEIGHTH, THIRTYNINTH, FORTIETH, FORTYFIRST, FORTYSECOND, FORTYTHIRD
Controllable:No
Description:Order of the FE shape function to use for this variable (additional orders not listed here are allowed, depending on the family).
verboseFalseFlag to facilitate debugging a Physics
Default:False
C++ Type:bool
Controllable:No
Description:Flag to facilitate debugging a Physics

Optional Parameters

active__all__ If specified only the blocks named will be visited and made active
Default:__all__
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified only the blocks named will be visited and made active
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.
inactiveIf specified blocks matching these identifiers will be skipped.
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified blocks matching these identifiers will be skipped.

Advanced Parameters

boundary_fluxesFunctors to compute the diffusive flux on each Neumann boundary'
C++ Type:std::vector<MooseFunctorName>
Controllable:No
Description:Functors to compute the diffusive flux on each Neumann boundary'
boundary_valuesFunctors to compute the diffusive flux on each Dirichlet boundary'
C++ Type:std::vector<MooseFunctorName>
Controllable:No
Description:Functors to compute the diffusive flux on each Dirichlet boundary'
dirichlet_boundariesBoundaries on which to apply a fixed value
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:Boundaries on which to apply a fixed value
neumann_boundariesBoundaries on which to apply a diffusive flux
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:Boundaries on which to apply a diffusive flux

Boundary Conditions Parameters

initial_from_file_timestepLATESTGives the time step number (or "LATEST") for which to read the Exodus solution
Default:LATEST
C++ Type:std::string
Controllable:No
Description:Gives the time step number (or "LATEST") for which to read the Exodus solution
initialize_variables_from_mesh_fileFalseDetermines if the variables that are added by the action are initializedfrom the mesh file (only for Exodus format)
Default:False
C++ Type:bool
Controllable:No
Description:Determines if the variables that are added by the action are initializedfrom the mesh file (only for Exodus format)