ArrayDiffusion

Description

This array kernel implements the following piece of a weak form:

$var element = document.getElementById("moose-equation-c629e545-d8b6-4f8d-bb8a-9a8abfeb581d");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}),", element, {displayMode:true,throwOnError:false});$ which is expanded as $var element = document.getElementById("moose-equation-3399dc61-b341-4cd7-b02d-a29a95c1ea34");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{p=1}^N w_p \\sum_{q=1}^N (\\nabla u_p^\\ast, D_{p,q} \\nabla u_q),", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-aa01c20a-e08d-4baa-ae78-2a0622d95222");katex.render("\\vec{u}^\\ast", element, {displayMode:false,throwOnError:false});$ is the test function, $var element = document.getElementById("moose-equation-7be11397-c0f9-48d3-bffc-3536e578ab77");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false});$ is the finite element solution and $var element = document.getElementById("moose-equation-3cb5736e-0a1a-4122-9041-248266cf0cdd");katex.render("\\mathbf{D}", element, {displayMode:false,throwOnError:false});$ is the diffusion coefficients. $var element = document.getElementById("moose-equation-e5011e51-affd-4f74-b20a-2546a7c90bdf");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false});$ is an array variable that has $var element = document.getElementById("moose-equation-ea154922-27ba-46ee-8cdd-8b94b6f59a8c");katex.render("N", element, {displayMode:false,throwOnError:false});$ number of components. $var element = document.getElementById("moose-equation-585fba13-5a78-4ae4-a1c4-adfe832042fc");katex.render("w_p, p=1,\\cdots,N", element, {displayMode:false,throwOnError:false});$ is the scalings of all components of the array variable. The size of the vector test function $var element = document.getElementById("moose-equation-58e149d4-f5fa-4ea2-b8e3-317a4d0fa6c2");katex.render("\\vec{u}^\\ast", element, {displayMode:false,throwOnError:false});$ is the same as the size of $var element = document.getElementById("moose-equation-0bd848c3-55b0-4744-b01d-516676e1f20b");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false});$ . The kernel can be further spelled out as $(1)var element = document.getElementById("moose-equation-a673a523-41b3-4682-b35f-954657005b3e");katex.render("\\sum_{p=1}^N w_p \\sum_{q=1}^N (\\nabla u_p^\\ast, D_{p,q} \\nabla u_q) = \\sum_{p=1}^N w_p \\sum_{q=1}^N \\sum_{e} \\int_e D_{p,q}(x) \\nabla u_p^\\ast (x) \\cdot \\nabla u_q(x)\\,dx = \\sum_{p=1}^N w_p \\sum_{q=1}^N \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{j=1}^{N_{\\text{dof}}} u_{p,i}^\\ast u_{q,j} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} D_{p,q,\\text{qp}} \\nabla b_{i,\\text{qp}} \\cdot \\nabla b_{j,\\text{qp}}, ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-f9ded0c3-2299-4d3c-8a43-2427e12937bd");katex.render("e", element, {displayMode:false,throwOnError:false});$ denotes elements of the mesh; $var element = document.getElementById("moose-equation-6e9061c0-6305-4b69-8497-ddb8dd10df4b");katex.render("N_\\text{dof}", element, {displayMode:false,throwOnError:false});$ is the number of shape functions on the local element; $var element = document.getElementById("moose-equation-1f77943a-09b4-4364-88ea-0518216fce94");katex.render("N_\\text{qp}", element, {displayMode:false,throwOnError:false});$ is the number of quadrature points for doing the spatial integration over an element. $var element = document.getElementById("moose-equation-30985c35-310a-4419-8129-b6b6c2c6eee6");katex.render("u_{p,i}^\\ast", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-13d5c21b-5fe3-4fb2-8e4d-876fa428fec4");katex.render("u_{p,i}^\\ast", element, {displayMode:false,throwOnError:false});$ are the expansion coefficients for the test function and the solution respectively, often referred as degrees of freedom. Subscript $var element = document.getElementById("moose-equation-9709fb32-56da-4841-a8eb-0186b362ccd9");katex.render("\\text{qp}", element, {displayMode:false,throwOnError:false});$ indicates that the associated quantities are evaluated on a quadrature point. $var element = document.getElementById("moose-equation-037c5231-84ca-46b8-877b-ec228af90c1e");katex.render("(|J|w)_\\text{qp}", element, {displayMode:false,throwOnError:false});$ is the determinant of Jacobian, that transform a physical element to a reference element where the shape functions $var element = document.getElementById("moose-equation-1d91ef15-9b4d-45e6-85d7-622b8e86f4cd");katex.render("b_i(\\hat{x}),i=1,\\cdots,N_\\text{dof}", element, {displayMode:false,throwOnError:false});$ are defined, times local quadrature weighting. It is noted that the test functions of all components are identical because all components of an array variable share the same finite element family and order.

We can rearrange the fully expanded Eq. (1) into $var element = document.getElementById("moose-equation-5c7a2a72-64ee-4aa9-b79d-676c4344cef1");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} \\vec{w}_p\\vec{u}_i^\\ast \\underline{\\mathbf{D}_{\\text{qp}} \\nabla \\vec{u}_{\\text{qp}} \\cdot \\nabla b_{i,\\text{qp}}},", element, {displayMode:true,throwOnError:false});$ where the underlined term is the array provided by ArrayDiffusion::computeQpResidual. The element, shape function and quadrature point summations are taken care of by MOOSE. It is noted that since test functions are arbitrary, $var element = document.getElementById("moose-equation-64e74194-4842-43d2-b6e9-805dbd6700ff");katex.render("\\vec{u}_i^\\ast", element, {displayMode:false,throwOnError:false});$ can be viewed as an index indicator with which the local residual goes into the global residual vector. Note that $var element = document.getElementById("moose-equation-6bda74d9-f634-4408-8753-c83fb0f2c0fe");katex.render("\\vec{a}\\vec{b}", element, {displayMode:false,throwOnError:false});$ represents element-wise multiplication, i.e. $var element = document.getElementById("moose-equation-4cde09a3-444a-4cbe-9996-06c71588e8c1");katex.render("\\vec{a}\\vec{b}", element, {displayMode:false,throwOnError:false});$ is equal to vector whose ith element is $var element = document.getElementById("moose-equation-252ea1aa-f10b-4555-9575-0578e3e86373");katex.render("a_i \\times b_i", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-f25c052b-b1fc-425f-927a-b4c97076183c");katex.render("a", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-518d7817-0ac3-44ea-840c-beb6b8a2b23e");katex.render("b", element, {displayMode:false,throwOnError:false});$ are two generic vectors.

In general, the diffusion coefficient $var element = document.getElementById("moose-equation-94e01ba7-b105-4af5-9886-85094e97e207");katex.render("\\mathbf{D}", element, {displayMode:false,throwOnError:false});$ is a square matrix with the size of the number of components. When it is a diagonal matrix, it can be represented by an array. In such a case, the components are not coupled with this array diffusion kernel. If all elements of the diffusion coefficient vector are the same, we can use a scalar diffusion coefficient. Thus this kernel gives users an option to set the type of diffusion coefficient with a parameter named as diffusion_coefficient_type. Users can set it to scalar, array or full corresponding to scalar, diagonal matrix and full matrix respectively. Its default value is array.

With some further transformation, the kernel becomes $var element = document.getElementById("moose-equation-bc51d90f-e4aa-4011-b593-7d405eb3c5c0");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{j=1}^{N_{\\text{dof}}} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} \\vec{w}_p\\vec{u}_i^\\ast \\underline{\\mathbf{D}_{\\text{qp}} \\nabla b_{j,\\text{qp}} \\cdot \\nabla b_{i,\\text{qp}}} \\vec{u}_j,", element, {displayMode:true,throwOnError:false});$ where the underlined part is the local Jacobian evaluated by ArrayDiffusion::computeQpJacobian and ArrayDiffusion::computeQpOffDiagJacobian.

Example Input Syntax

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
[]

Input Parameters

diffusion_coefficientThe name of the diffusivity, can be scalar, vector, or matrix material property.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The name of the diffusivity, can be scalar, vector, or matrix material property.
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.
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.
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>
Unit:(no unit assumed)
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
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
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>
Unit:(no unit assumed)
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
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

(moose/framework/src/kernels/ArrayDiffusion.C)

// 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

#include "ArrayDiffusion.h"

registerMooseObject("MooseApp", ArrayDiffusion);

InputParameters
ArrayDiffusion::validParams()
{
  InputParameters params = ArrayKernel::validParams();
  params.addRequiredParam<MaterialPropertyName>(
      "diffusion_coefficient",
      "The name of the diffusivity, can be scalar, vector, or matrix material property.");
  params.addClassDescription(
      "The array Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak "
      "form of $(\\nabla \\phi_i, \\nabla u_h)$.");
  return params;
}

ArrayDiffusion::ArrayDiffusion(const InputParameters & parameters)
  : ArrayKernel(parameters),
    _d(hasMaterialProperty<Real>("diffusion_coefficient")
           ? &getMaterialProperty<Real>("diffusion_coefficient")
           : nullptr),
    _d_array(hasMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 ? &getMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 : nullptr),
    _d_2d_array(hasMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    ? &getMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    : nullptr)
{
  if (!_d && !_d_array && !_d_2d_array)
  {
    MaterialPropertyName mat = getParam<MaterialPropertyName>("diffusion_coefficient");
    mooseError("Property " + mat + " is of unsupported type for ArrayDiffusion");
  }
}

void
ArrayDiffusion::initQpResidual()
{
  if (_d_array)
  {
    mooseAssert((*_d_array)[_qp].size() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
  else if (_d_2d_array)
  {
    mooseAssert((*_d_2d_array)[_qp].cols() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
    mooseAssert((*_d_2d_array)[_qp].rows() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
}

void
ArrayDiffusion::computeQpResidual(RealEigenVector & residual)
{
  // WARNING: the noalias() syntax is an Eigen optimization tactic, it avoids creating
  // a temporary object for the matrix multiplication on the right-hand-side. However,
  // it should be used with caution because it could cause unintended results,
  // developers should NOT use it if the vector on the left-hand-side appears on the
  // right-hand-side, for instance:
  //   vector = matrix * vector;
  // See http://eigen.tuxfamily.org/dox/group__TopicAliasing.html for more details.
  if (_d)
    residual.noalias() = (*_d)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else if (_d_array)
    residual.noalias() = (*_d_array)[_qp].asDiagonal() * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else
    residual.noalias() = (*_d_2d_array)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
}

RealEigenVector
ArrayDiffusion::computeQpJacobian()
{
  if (_d)
    return RealEigenVector::Constant(_var.count(),
                                     _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d)[_qp]);
  else if (_d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_array)[_qp];
  else
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp].diagonal();
}

RealEigenMatrix
ArrayDiffusion::computeQpOffDiagJacobian(const MooseVariableFEBase & jvar)
{
  if (jvar.number() == _var.number() && _d_2d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp];
  else
    return ArrayKernel::computeQpOffDiagJacobian(jvar);
}

(moose/framework/src/kernels/ArrayDiffusion.C)

// 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

#include "ArrayDiffusion.h"

registerMooseObject("MooseApp", ArrayDiffusion);

InputParameters
ArrayDiffusion::validParams()
{
  InputParameters params = ArrayKernel::validParams();
  params.addRequiredParam<MaterialPropertyName>(
      "diffusion_coefficient",
      "The name of the diffusivity, can be scalar, vector, or matrix material property.");
  params.addClassDescription(
      "The array Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak "
      "form of $(\\nabla \\phi_i, \\nabla u_h)$.");
  return params;
}

ArrayDiffusion::ArrayDiffusion(const InputParameters & parameters)
  : ArrayKernel(parameters),
    _d(hasMaterialProperty<Real>("diffusion_coefficient")
           ? &getMaterialProperty<Real>("diffusion_coefficient")
           : nullptr),
    _d_array(hasMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 ? &getMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 : nullptr),
    _d_2d_array(hasMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    ? &getMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    : nullptr)
{
  if (!_d && !_d_array && !_d_2d_array)
  {
    MaterialPropertyName mat = getParam<MaterialPropertyName>("diffusion_coefficient");
    mooseError("Property " + mat + " is of unsupported type for ArrayDiffusion");
  }
}

void
ArrayDiffusion::initQpResidual()
{
  if (_d_array)
  {
    mooseAssert((*_d_array)[_qp].size() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
  else if (_d_2d_array)
  {
    mooseAssert((*_d_2d_array)[_qp].cols() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
    mooseAssert((*_d_2d_array)[_qp].rows() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
}

void
ArrayDiffusion::computeQpResidual(RealEigenVector & residual)
{
  // WARNING: the noalias() syntax is an Eigen optimization tactic, it avoids creating
  // a temporary object for the matrix multiplication on the right-hand-side. However,
  // it should be used with caution because it could cause unintended results,
  // developers should NOT use it if the vector on the left-hand-side appears on the
  // right-hand-side, for instance:
  //   vector = matrix * vector;
  // See http://eigen.tuxfamily.org/dox/group__TopicAliasing.html for more details.
  if (_d)
    residual.noalias() = (*_d)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else if (_d_array)
    residual.noalias() = (*_d_array)[_qp].asDiagonal() * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else
    residual.noalias() = (*_d_2d_array)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
}

RealEigenVector
ArrayDiffusion::computeQpJacobian()
{
  if (_d)
    return RealEigenVector::Constant(_var.count(),
                                     _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d)[_qp]);
  else if (_d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_array)[_qp];
  else
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp].diagonal();
}

RealEigenMatrix
ArrayDiffusion::computeQpOffDiagJacobian(const MooseVariableFEBase & jvar)
{
  if (jvar.number() == _var.number() && _d_2d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp];
  else
    return ArrayKernel::computeQpOffDiagJacobian(jvar);
}

(moose/framework/src/kernels/ArrayDiffusion.C)

// 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

#include "ArrayDiffusion.h"

registerMooseObject("MooseApp", ArrayDiffusion);

InputParameters
ArrayDiffusion::validParams()
{
  InputParameters params = ArrayKernel::validParams();
  params.addRequiredParam<MaterialPropertyName>(
      "diffusion_coefficient",
      "The name of the diffusivity, can be scalar, vector, or matrix material property.");
  params.addClassDescription(
      "The array Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak "
      "form of $(\\nabla \\phi_i, \\nabla u_h)$.");
  return params;
}

ArrayDiffusion::ArrayDiffusion(const InputParameters & parameters)
  : ArrayKernel(parameters),
    _d(hasMaterialProperty<Real>("diffusion_coefficient")
           ? &getMaterialProperty<Real>("diffusion_coefficient")
           : nullptr),
    _d_array(hasMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 ? &getMaterialProperty<RealEigenVector>("diffusion_coefficient")
                 : nullptr),
    _d_2d_array(hasMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    ? &getMaterialProperty<RealEigenMatrix>("diffusion_coefficient")
                    : nullptr)
{
  if (!_d && !_d_array && !_d_2d_array)
  {
    MaterialPropertyName mat = getParam<MaterialPropertyName>("diffusion_coefficient");
    mooseError("Property " + mat + " is of unsupported type for ArrayDiffusion");
  }
}

void
ArrayDiffusion::initQpResidual()
{
  if (_d_array)
  {
    mooseAssert((*_d_array)[_qp].size() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
  else if (_d_2d_array)
  {
    mooseAssert((*_d_2d_array)[_qp].cols() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
    mooseAssert((*_d_2d_array)[_qp].rows() == _var.count(),
                "diffusion_coefficient size is inconsistent with the number of components of array "
                "variable");
  }
}

void
ArrayDiffusion::computeQpResidual(RealEigenVector & residual)
{
  // WARNING: the noalias() syntax is an Eigen optimization tactic, it avoids creating
  // a temporary object for the matrix multiplication on the right-hand-side. However,
  // it should be used with caution because it could cause unintended results,
  // developers should NOT use it if the vector on the left-hand-side appears on the
  // right-hand-side, for instance:
  //   vector = matrix * vector;
  // See http://eigen.tuxfamily.org/dox/group__TopicAliasing.html for more details.
  if (_d)
    residual.noalias() = (*_d)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else if (_d_array)
    residual.noalias() = (*_d_array)[_qp].asDiagonal() * _grad_u[_qp] * _array_grad_test[_i][_qp];
  else
    residual.noalias() = (*_d_2d_array)[_qp] * _grad_u[_qp] * _array_grad_test[_i][_qp];
}

RealEigenVector
ArrayDiffusion::computeQpJacobian()
{
  if (_d)
    return RealEigenVector::Constant(_var.count(),
                                     _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d)[_qp]);
  else if (_d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_array)[_qp];
  else
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp].diagonal();
}

RealEigenMatrix
ArrayDiffusion::computeQpOffDiagJacobian(const MooseVariableFEBase & jvar)
{
  if (jvar.number() == _var.number() && _d_2d_array)
    return _grad_phi[_j][_qp] * _grad_test[_i][_qp] * (*_d_2d_array)[_qp];
  else
    return ArrayKernel::computeQpOffDiagJacobian(jvar);
}

(moose/test/tests/kernels/array_kernels/array_diffusion_test.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 4
  ny = 4
[]

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
    scaling = '.9 .9'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
[]

[BCs]
  [left]
    type = ArrayDirichletBC
    variable = u
    boundary = 1
    values = '0 0'
  []

  [right]
    type = ArrayDirichletBC
    variable = u
    boundary = 2
    values = '1 2'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
[]

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

[Outputs]
  exodus = true
[]

Description
Example Input Syntax
Input Parameters