AnisotropicDiffusion

Description

The AnisotropicDiffusion kernel implements anisotropic diffusion term on a domain ( $var element = document.getElementById("moose-equation-e546ed9f-23ed-4fa5-8c0a-c5808ba40c2d");katex.render("\\Omega", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) given in its strong form as

$var element = document.getElementById("moose-equation-ccd5f5ec-cec7-45a3-bcfe-6235454eb3b5");katex.render("\\nabla\\cdot -\\widetilde{k} \\nabla u = 0 \\in \\Omega,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-4b557cb4-b2f0-43e8-ba05-081af7bd81e4");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the anisotropic diffusion coefficient. Diffusion is anisotropic if the diffusion rate varies with direction. The corresponding weak form, using inner-product notation, is given by

$var element = document.getElementById("moose-equation-1d696e0c-9966-4a0b-878f-ea45f1ee5583");katex.render("R_i(u_h) = \\underbrace{(\\nabla \\psi_i, \\widetilde{k} \\nabla u_h)}_{\\textrm{AnisotropicDiffusion}} - \\langle\\psi_i, \\widetilde{k} \\nabla u_h \\cdot \\vec{n}\\rangle\\quad \\forall \\phi_i,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where the first term denotes the inner product over the domain volume, the latter term denotes the outward diffusion flux over the volume boundary ( $var element = document.getElementById("moose-equation-d4095992-9f41-4214-a1f9-8bf5f266ac26");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ), $var element = document.getElementById("moose-equation-6baff8ff-9461-49b5-ab47-21c40aecc229");katex.render("\\psi_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are the test functions, and $var element = document.getElementById("moose-equation-597982a7-7232-4f51-8b52-acee7020881e");katex.render("u_h \\in \\mathcal{S}^h", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ 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 $var element = document.getElementById("moose-equation-cdf7d043-01ae-462c-b597-9f525d1b1856");katex.render("\\frac{\\partial R_i(u_h)}{\\partial u_j} = (\\nabla \\phi_j, \\widetilde{k} \\nabla u_h).", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

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 AnisotropicDiffusion syntax in a steady-state anisotropic 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 anisotropic diffusion coefficient $var element = document.getElementById("moose-equation-efbe1207-7f13-45be-9df9-f0272896ba6e");katex.render("\\widetilde{k}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a three-dimensional tensor supplied through a string with nine space separated real values. The entries correspond to $var element = document.getElementById("moose-equation-13c39a3c-ba1b-4958-aef6-cba548903b87");katex.render("xx", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-903db6fa-736b-44bb-a8c4-318cd4045bb8");katex.render("xy", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-64edbfe1-152e-46b4-98a0-e192869ccf23");katex.render("xz", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-fb679324-fe29-4bd0-8a39-92f82abb200f");katex.render("yx", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-14d4d4d5-ece7-435f-9941-fc878e3d6c0f");katex.render("yy", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-134d6aea-51a5-40d3-a6ef-9b0324b79239");katex.render("yz", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-3af78096-d385-4b4e-ac25-0aef7d7d41b7");katex.render("zx", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-f49ebaca-e33f-408d-92ab-d57bae3ca8ea");katex.render("zy", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and $var element = document.getElementById("moose-equation-5bb10b67-a2f1-46c7-b986-daa6298287cc");katex.render("zz", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , respectively. Also, this problem is 2-dimensional so the corresponding $var element = document.getElementById("moose-equation-238fdba0-3994-4a82-b1d2-592b94970fe4");katex.render("zx", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-8a86b44c-89f2-4edb-a1e5-fbc774c40634");katex.render("zy", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and $var element = document.getElementById("moose-equation-b9fbbb32-6a89-4883-97cf-a67adc5b7edb");katex.render("zz", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ terms are zero.

Input Parameters

tensor_coeffThe Tensor to multiply the Diffusion operator by
C++ Type:libMesh::TensorValue<double>
Controllable:No
Description:The Tensor to multiply the Diffusion operator by
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
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>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
displacementsThe displacements
C++ Type:std::vector<VariableName>
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
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.
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
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>
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_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
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>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
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<AuxVariableName>
Controllable:No
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.)
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
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<AuxVariableName>
Controllable:No
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.)
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

Input Files

(test/tests/kernels/anisotropic_diffusion/aniso_diffusion.i)
(modules/misc/test/tests/fracture_flow/single.i)
(modules/porous_flow/examples/flow_through_fractured_media/diffusion.i)
(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/no_multiapp.i)
(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/fracture_app_dirac.i)
(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/matrix_app_dirac.i)
(test/tests/misc/check_error/missing_required_parameter_moose_obj_test.i)
(test/tests/fvkernels/fv_anisotropic_diffusion/fv_anisotropic_diffusion.i)
(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/matrix_app_nonconforming.i)
(modules/porous_flow/examples/multiapp_fracture_flow/single_fracture_heat_transfer/matrix_app.i)

Child Objects

(modules/geochemistry/include/kernels/GeochemistryDispersion.h)

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

[Mesh]
  file = mixed_block.e
  uniform_refine=3
[]

[Functions]
  [./top_bc]
    type = ParsedFunction
    expression = 'x'
  [../]
[]

[Variables]
  active = 'u'

  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

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

[BCs]
  active = 'lower_left top'

  [./lower_left]
    type = DirichletBC
    variable = u
    boundary = '1 4'
    value = 1
  [../]

  [./top]
    type = FunctionNeumannBC
    variable = u
    boundary = 3
    function = top_bc
  [../]

  [./right]
    type = NeumannBC
    variable = u
    boundary = 2
  [../]
[]

[Executioner]
  type = Steady

  solve_type = 'PJFNK'
[]

[Outputs]
  exodus = true
[]

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

[Mesh]
  file = mixed_block.e
  uniform_refine=3
[]

[Functions]
  [./top_bc]
    type = ParsedFunction
    expression = 'x'
  [../]
[]

[Variables]
  active = 'u'

  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

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

[BCs]
  active = 'lower_left top'

  [./lower_left]
    type = DirichletBC
    variable = u
    boundary = '1 4'
    value = 1
  [../]

  [./top]
    type = FunctionNeumannBC
    variable = u
    boundary = 3
    function = top_bc
  [../]

  [./right]
    type = NeumannBC
    variable = u
    boundary = 2
  [../]
[]

[Executioner]
  type = Steady

  solve_type = 'PJFNK'
[]

[Outputs]
  exodus = true
[]

(modules/misc/test/tests/fracture_flow/single.i)

# Models fluid advecting down a single fracture sitting at x=0, and 0<=y<=3.
#
[Mesh]
  type = FileMesh
  file = 'single.e'
[]

[Variables]
  [./u]
  [../]
[]

[ICs]
  [./u_init]
    type = ConstantIC
    variable = u
    value = 0
  [../]
[]

[BCs]
  [./inj]
    type = DirichletBC
    boundary = 1
    variable = u
    value = 1
  [../]
[]

[Kernels]
  [./matrix_dt]
    type = CoefTimeDerivative
    variable = u
    Coefficient = 0.2  # matrix porosity
    block = 1
  [../]
  [./matrix_diff]
    type = AnisotropicDiffusion
    variable = u
    block = 1
    tensor_coeff = '0.002 0 0   0 0 0   0 0 0'  # matrix porosity * matrix diffusivity
  [../]
  [./fracture_dt]
    type = CoefTimeDerivative
    variable = u
    Coefficient = 0.1  # fracture half-aperture * fracture porosity
    block = 2
  [../]
  [./fracture_advect]
    type = Convection
    variable = u
    block = 2
    velocity = '0 0.08 0' # fracture half-aperture * velocity in fracture
  [../]
[]

[Preconditioning]
  [./standard]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  dt = 2e-1
  end_time = 1.0
  solve_type = Newton
  nl_rel_tol = 1E-12
[]

[Outputs]
  exodus = true
[]

(modules/porous_flow/examples/flow_through_fractured_media/diffusion.i)

[Mesh]
  file = diffusion_1.e # or diffusion_5.e or diffusion_fine.e
[]

[Variables]
  [T]
  []
[]

[BCs]
  [left]
    type = DirichletBC
    boundary = 2
    variable = T
    value = 1
  []
[]

[Kernels]
  [dot]
    type = TimeDerivative
    variable = T
  []
  [fracture_diffusion]
    type = AnisotropicDiffusion
    block = 1
    tensor_coeff = '1 0 0  0 1 0  0 0 1'
    variable = T
  []
  [matrix_diffusion]
    type = AnisotropicDiffusion
    block = '2 3'
    tensor_coeff = '0 0 0  0 0 0  0 0 0'
    variable = T
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 10
  end_time = 100
  nl_abs_tol = 1E-13
  nl_rel_tol = 1E-12
[]

[Outputs]
  print_linear_residuals = false
  exodus = true
[]

(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/no_multiapp.i)

# A fracture, which is a 1D line of elements, is embedded in a matrix, which is a 2D surface of elements.
# The meshes conform: all fracture nodes are also matrix nodes (the fracture elements are sides of matrix elements).
# The overall mesh has two blocks, named "matrix" and "fracture".
#
# Two variables are defined:
# - frac_T, which is the temperature inside the fracture;
# - matrix_T, which is the temperature in the matrix.
# frac_T is governed by a diffusion equation along the 1D fracture.
# matrix_T is governed by a diffusion equation in the 2D matrix, with small diffusion coefficient.
# Heat is exchanged between the two systems via a heat-transfer coefficient, defined on the fracture subdomain, using two PorousFlowHeatMassTransfer Kernels
#
# If the mesh is too coarse, overshoots and undershoots in matrix_T can be observed.
[Mesh]
  [generate]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    xmin = 0
    xmax = 10.0
    ny = 20 # anything less than this produces over/under-shoots
    ymin = -2
    ymax = 2
  []
  [matrix_subdomain]
    type = RenameBlockGenerator
    input = generate
    old_block = 0
    new_block = matrix
  []
  [fracture_sideset]
    type = ParsedGenerateSideset
    input = matrix_subdomain
    combinatorial_geometry = 'y>-1E-6 & y<1E-6'
    normal = '0 1 0'
    new_sideset_name = fracture_sideset
  []
  [fracture_subdomain]
    type = LowerDBlockFromSidesetGenerator
    input = fracture_sideset
    new_block_id = 1
    new_block_name = fracture
    sidesets = fracture_sideset
  []
[]

[Variables]
  [frac_T]
    block = fracture
  []
  [matrix_T]
    # Needs to be defined on both blocks, so PorousFlowHeatMassTransfer works appropriately
    # Kernels for diffusion are on block=matrix only
  []
[]

[BCs]
  [frac_T]
    type = DirichletBC
    variable = frac_T
    boundary = left
    value = 1
  []
[]

[Kernels]
  [dot_frac_T]
    type = CoefTimeDerivative
    Coefficient = 1E-2
    variable = frac_T
    block = fracture
  []
  [fracture_diffusion]
    type = AnisotropicDiffusion
    variable = frac_T
    tensor_coeff = '1E-2 0 0 0 1E-2 0 0 0 1E-2'
    block = fracture
  []
  [toMatrix]
    type = PorousFlowHeatMassTransfer
    block = fracture
    variable = frac_T
    v = matrix_T
    transfer_coefficient = 0.02
  []
  [dot_matrix_T]
    type = TimeDerivative
    variable = matrix_T
    block = matrix
  []
  [matrix_diffusion]
    type = AnisotropicDiffusion
    variable = matrix_T
    tensor_coeff = '1E-3 0 0 0 1E-3 0 0 0 1E-3'
    block = matrix
  []
  [fromFracture]
    type = PorousFlowHeatMassTransfer
    block = fracture
    variable = matrix_T
    v = frac_T
    transfer_coefficient = 0.02
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 100
  end_time = 100
[]

[VectorPostprocessors]
  [frac_T]
    type = NodalValueSampler
    block = fracture
    outputs = frac_T
    sort_by = x
    variable = frac_T
  []
[]

[Outputs]
  print_linear_residuals = false
  exodus = false
  [frac_T]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/fracture_app_dirac.i)

# A fracture, which is a 1D line of elements, is embedded in a matrix, which is a 2D surface of elements.
#
# The heat equation governs temperature in the fracture and matrix system, and heat energy is transferred between the two using a MultiApp approach
[Mesh]
  [generate]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 20
    xmin = 0
    xmax = 10.0
  []
[]

[Variables]
  [frac_T]
  []
[]

[BCs]
  [frac_T]
    type = DirichletBC
    variable = frac_T
    boundary = left
    value = 1
  []
[]

[AuxVariables]
  [transferred_matrix_T]
  []
  [joules_per_s]
  []
[]

[Kernels]
  [dot_frac_T]
    type = CoefTimeDerivative
    Coefficient = 1E-2
    variable = frac_T
  []
  [fracture_diffusion]
    type = AnisotropicDiffusion
    variable = frac_T
    tensor_coeff = '1E-2 0 0 0 1E-2 0 0 0 1E-2'
  []
  [toMatrix]
    type = PorousFlowHeatMassTransfer
    variable = frac_T
    v = transferred_matrix_T
    transfer_coefficient = 0.02
    save_in = joules_per_s
  []
[]

[VectorPostprocessors]
  [heat_transfer_rate]
    type = NodalValueSampler
    outputs = none
    sort_by = id
    variable = joules_per_s
  []
  [frac_T]
    type = NodalValueSampler
    outputs = frac_T
    sort_by = x
    variable = frac_T
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 100
  end_time = 100
  nl_rel_tol = 1e-8
  petsc_options_iname = '-pc_type  -pc_factor_mat_solver_package'
  petsc_options_value = 'lu        superlu_dist'
[]

[Outputs]
  print_linear_residuals = false
  exodus = false
  [frac_T]
    type = CSV
    execute_on = final
  []
[]

(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/matrix_app_dirac.i)

# A fracture, which is a 1D line of elements, is embedded in a matrix, which is a 2D surface of elements.
# The meshes conform: all fracture nodes are also matrix nodes (the fracture elements are sides of matrix elements).
#
# The heat equation governs temperature in the fracture and matrix system, and heat energy is transferred between the two using a MultiApp approach
[Mesh]
  [generate]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    xmin = 0
    xmax = 10.0
    ny = 20 # anything less than this produces over/under-shoots
    ymin = -2
    ymax = 2
  []
[]

[Variables]
  [matrix_T]
  []
[]

[Kernels]
  [dot]
    type = TimeDerivative
    variable = matrix_T
  []
  [matrix_diffusion]
    type = AnisotropicDiffusion
    variable = matrix_T
    tensor_coeff = '1E-3 0 0 0 1E-3 0 0 0 1E-3'
  []
[]

[DiracKernels]
  [heat_from_fracture]
    type = ReporterPointSource
    variable = matrix_T
    value_name = heat_transfer_rate/transferred_joules_per_s
    x_coord_name = heat_transfer_rate/x
    y_coord_name = heat_transfer_rate/y
    z_coord_name = heat_transfer_rate/z
  []
[]

[VectorPostprocessors]
  [heat_transfer_rate]
    type = ConstantVectorPostprocessor
    vector_names = 'transferred_joules_per_s x y z'
    value = '0; 0; 0; 0'
    outputs = none
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 100
  end_time = 100
  nl_rel_tol = 1e-8
  petsc_options_iname = '-pc_type  -pc_factor_mat_solver_package'
  petsc_options_value = 'lu        superlu_dist'
[]


[Outputs]
  print_linear_residuals = false
  exodus = false
  csv=true
[]

[MultiApps]
  [fracture_app]
    type = TransientMultiApp
    input_files = fracture_app_dirac.i
    execute_on = TIMESTEP_BEGIN
  []
[]

[Transfers]
  [T_to_fracture]
    type = MultiAppGeometricInterpolationTransfer
    to_multi_app = fracture_app
    source_variable = matrix_T
    variable = transferred_matrix_T
  []
  [heat_from_fracture]
    type = MultiAppReporterTransfer
    from_multi_app = fracture_app
    from_reporters = 'heat_transfer_rate/joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
    to_reporters = 'heat_transfer_rate/transferred_joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
  []
[]

(test/tests/misc/check_error/missing_required_parameter_moose_obj_test.i)

[Mesh]
  [./square]
    type = GeneratedMeshGenerator
    nx = 2
    ny = 2
    dim = 2
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  # Test error message for missing required parameter
  [./diff]
    type = AnisotropicDiffusion
    variable = u
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = u
    boundary = 3
    value = 0
  [../]

  [./right]
    type = DirichletBC
    variable = u
    boundary = 1
    value = 1
  [../]
[]

[Executioner]
  type = Steady
  solve_type = 'PJFNK'
[]

[Outputs]
  file_base = out
[]

(test/tests/fvkernels/fv_anisotropic_diffusion/fv_anisotropic_diffusion.i)

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 2
    dx = '10 10'
    ix = '2 2'
    dy = '20'
    iy = '4'
    subdomain_id = '1 2'
  []
[]

[Variables]
  [v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  []

  [u]
    order = FIRST
    family = LAGRANGE
  []
[]

[Kernels]
  [fem_diff1]
    type = AnisotropicDiffusion
    variable = u
    tensor_coeff = '1 0 0
                    0 10 0
                    0 0 0'
    block = 1
  []

  [fem_diff2]
    type = AnisotropicDiffusion
    variable = u
    tensor_coeff = '10 0 0
                    0 10 0
                    0 0 0'
    block = 2
  []
[]

[BCs]
  [fem_left_bottom]
    type = NeumannBC
    variable = u
    boundary = 'left bottom'
    value = 1
  []
  [fem_top_right]
    type = DirichletBC
    variable = u
    boundary = 'right top'
    value = 0
  []
[]

[FVKernels]
  [diff]
    type = FVAnisotropicDiffusion
    variable = v
    coeff = coeff
  []
[]

[FVBCs]
  [left_bottom]
    type = FVNeumannBC
    variable = v
    boundary = 'left bottom'
    value = 1
  []
  [top_right]
    type = FVDirichletBC
    variable = v
    boundary = 'right top'
    value = 0
  []
[]

[Materials]
  [diff1]
    type = ADGenericVectorFunctorMaterial
    prop_names = 'coeff'
    prop_values = '1 10 1'
    block = 1
  []

  [diff2]
    type = ADGenericVectorFunctorMaterial
    prop_names = 'coeff'
    prop_values = '10 10 1'
    block = 2
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

(modules/porous_flow/examples/multiapp_fracture_flow/fracture_diffusion/matrix_app_nonconforming.i)

# A fracture, which is a 1D line of elements, is embedded in a matrix, which is a 2D surface of elements.
# The meshes conform: all fracture nodes are also matrix nodes (the fracture elements are sides of matrix elements).
#
# The heat equation governs temperature in the fracture and matrix system, and heat energy is transferred between the two using a MultiApp approach
[Mesh]
  [generate]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    xmin = 0
    xmax = 10.0
    ny = 20
    ymin = -1.9
    ymax = 2.1
  []
[]

[Variables]
  [matrix_T]
  []
[]

[Kernels]
  [dot]
    type = TimeDerivative
    variable = matrix_T
  []
  [matrix_diffusion]
    type = AnisotropicDiffusion
    variable = matrix_T
    tensor_coeff = '1E-3 0 0 0 1E-3 0 0 0 1E-3'
  []
[]

[DiracKernels]
  [heat_from_fracture]
    type = ReporterPointSource
    variable = matrix_T
    value_name = heat_transfer_rate/transferred_joules_per_s
    x_coord_name = heat_transfer_rate/x
    y_coord_name = heat_transfer_rate/y
    z_coord_name = heat_transfer_rate/z
  []
[]

[VectorPostprocessors]
  [heat_transfer_rate]
    type = ConstantVectorPostprocessor
    vector_names = 'transferred_joules_per_s x y z'
    value = '0; 0; 0; 0'
    outputs = none
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 100
  end_time = 100
[]


[Outputs]
  print_linear_residuals = false
  exodus = false
[]

[MultiApps]
  [fracture_app]
    type = TransientMultiApp
    input_files = fracture_app_dirac.i
    cli_args = 'Kernels/toMatrix/transfer_coefficient=0.01'
    execute_on = TIMESTEP_BEGIN
  []
[]

[Transfers]
  [T_to_fracture]
    type = MultiAppGeometricInterpolationTransfer
    to_multi_app = fracture_app
    source_variable = matrix_T
    variable = transferred_matrix_T
  []
  [heat_from_fracture]
    type = MultiAppReporterTransfer
    from_multi_app = fracture_app
    from_reporters = 'heat_transfer_rate/joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
    to_reporters = 'heat_transfer_rate/transferred_joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
  []
[]

(modules/porous_flow/examples/multiapp_fracture_flow/single_fracture_heat_transfer/matrix_app.i)

# Matrix physics, which is just heat conduction.  Heat energy comes from the fracture App
[Mesh]
  [generate]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 20
    xmin = 0
    xmax = 100.0
    ny = 9
    ymin = -9
    ymax = 9
  []
[]

[Variables]
  [matrix_T]
    initial_condition = 40 # degC
  []
[]

[Kernels]
  [dot]
    type = CoefTimeDerivative
    variable = matrix_T
    Coefficient = 1E5
  []
  [matrix_diffusion]
    type = AnisotropicDiffusion
    variable = matrix_T
    tensor_coeff = '1 0 0 0 1 0 0 0 1'
  []
[]

[DiracKernels]
  [heat_from_fracture]
    type = ReporterPointSource
    variable = matrix_T
    value_name = heat_transfer_rate/transferred_joules_per_s
    x_coord_name = heat_transfer_rate/x
    y_coord_name = heat_transfer_rate/y
    z_coord_name = heat_transfer_rate/z
  []
[]

[VectorPostprocessors]
  [heat_transfer_rate]
    type = ConstantVectorPostprocessor
    vector_names = 'transferred_joules_per_s x y z'
    value = '0; 0; 0; 0'
    outputs = none
  []
[]

[Preconditioning]
  [entire_jacobian]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  dt = 1
  end_time = 100
  nl_abs_tol = 1E-3
[]


[Outputs]
  print_linear_residuals = false
  exodus = true
  csv=true
[]

[MultiApps]
  [fracture_app]
    type = TransientMultiApp
    input_files = fracture_app.i
    execute_on = TIMESTEP_BEGIN
  []
[]

[Transfers]
  [T_to_fracture]
    type = MultiAppGeometricInterpolationTransfer
    to_multi_app = fracture_app
    source_variable = matrix_T
    variable = transferred_matrix_T
  []
  [heat_from_fracture]
    type = MultiAppReporterTransfer
    from_multi_app = fracture_app
    from_reporters = 'heat_transfer_rate/joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
    to_reporters = 'heat_transfer_rate/transferred_joules_per_s heat_transfer_rate/x heat_transfer_rate/y heat_transfer_rate/z'
  []
[]

(modules/geochemistry/include/kernels/GeochemistryDispersion.h)

// This file is part of the MOOSE framework
// https://www.mooseframework.org
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "AnisotropicDiffusion.h"

/**
 * Kernel describing grad(porosity * dispersion * grad(concentration)), where porosity is an
 * AuxVariable, dispersion is the hydrodynamic dispersion tensor (input by user as tensor_coeff),
 * and concentration is the variable for this Kernel
 */
class GeochemistryDispersion : public AnisotropicDiffusion
{
public:
  static InputParameters validParams();

  GeochemistryDispersion(const InputParameters & parameters);

protected:
  virtual Real computeQpResidual() override;
  virtual Real computeQpJacobian() override;

private:
  const VariableValue & _porosity;
};

Description
Example Syntax
Input Parameters
Input Files
Child Objects