ArrayDiffusion

Description

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

$var element = document.getElementById("moose-equation-c8ed57c2-2143-4f7b-83d7-4ae90de4c136");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}),", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ which is expanded as $var element = document.getElementById("moose-equation-e63ea4e1-476e-4f78-bda6-c240095d3e6b");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{p=1}^N w_p \\sum_{q=1}^N (\\nabla u_p^\\ast, D_{p,q} \\nabla u_q),", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-c7ec1801-ab69-4eda-aa1f-4d4089626c09");katex.render("\\vec{u}^\\ast", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the test function, $var element = document.getElementById("moose-equation-f1016fdf-7c64-46e8-ae8c-53ef0b862122");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the finite element solution and $var element = document.getElementById("moose-equation-6fcd424a-8767-42f1-a02a-9fc51dc1e994");katex.render("\\mathbf{D}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the diffusion coefficients. $var element = document.getElementById("moose-equation-6bb163f7-1e96-458a-af96-187d4d58ba7e");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is an array variable that has $var element = document.getElementById("moose-equation-a92811ff-ef08-48dd-aeca-caddf1b42bca");katex.render("N", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ number of components. $var element = document.getElementById("moose-equation-f12aa00e-eb4e-4daa-bab7-4d5e080f5366");katex.render("w_p, p=1,\\cdots,N", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the scalings of all components of the array variable. The size of the vector test function $var element = document.getElementById("moose-equation-8b0c0ea1-818c-4bcf-89c4-5d1d7e5fa6af");katex.render("\\vec{u}^\\ast", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the same as the size of $var element = document.getElementById("moose-equation-7f10fc14-0314-4ff2-b599-7a18837bd07c");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . The kernel can be furthur spelled out as $(1)var element = document.getElementById("moose-equation-8edc03e1-0c67-48ae-b438-4744a31adf24");katex.render("\\sum_{p=1}^N w_p \\sum_{q=1}^N (\\nabla u_p^\\ast, D_{p,q} \\nabla u_q) = \\sum_{p=1}^N w_p \\sum_{q=1}^N \\sum_{e} \\int_e D_{p,q}(x) \\nabla u_p^\\ast (x) \\cdot \\nabla u_q(x)\\,dx = \\sum_{p=1}^N w_p \\sum_{q=1}^N \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{j=1}^{N_{\\text{dof}}} u_{p,i}^\\ast u_{q,j} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} D_{p,q,\\text{qp}} \\nabla b_{i,\\text{qp}} \\cdot \\nabla b_{j,\\text{qp}}, ", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-83e78cb0-67a7-4e4a-9703-558c71c7b745");katex.render("e", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ denotes elements of the mesh; $var element = document.getElementById("moose-equation-a15f5e7c-3d0d-40c2-98ab-bf040624b5ed");katex.render("N_\\text{dof}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the number of shape functions on the local element; $var element = document.getElementById("moose-equation-793e759b-6de2-47c6-9ca7-e5c4f8c6b453");katex.render("N_\\text{qp}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the number of quadrature points for doing the spatial integration over an element. $var element = document.getElementById("moose-equation-f6220875-ef59-40ce-98ea-5793a8407411");katex.render("u_{p,i}^\\ast", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-4182e434-3052-4fde-bf0f-bb1c2ea8c065");katex.render("u_{p,i}^\\ast", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are the expansion coefficients for the test function and the solution respectively, often referred as degrees of freedom. Subscript $var element = document.getElementById("moose-equation-1c1185cd-2b18-4fb5-b45c-80aa5ecb5885");katex.render("\\text{qp}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ indicates that the associated quantities are evaluated on a quadrature point. $var element = document.getElementById("moose-equation-ef19e78b-2cd7-4e79-926c-1b5c13dff143");katex.render("(|J|w)_\\text{qp}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the determinant of Jacobian, that transform a physical element to a reference element where the shape functions $var element = document.getElementById("moose-equation-bb54d9a5-e2af-43eb-ba82-5797b65b144a");katex.render("b_i(\\hat{x}),i=1,\\cdots,N_\\text{dof}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are 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 $var element = document.getElementById("moose-equation-a7ffc643-014e-47c2-8de4-c4f71ccac3b4");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} \\vec{w}_p\\vec{u}_i^\\ast \\underline{\\mathbf{D}_{\\text{qp}} \\nabla \\vec{u}_{\\text{qp}} \\cdot \\nabla b_{i,\\text{qp}}},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ 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, $var element = document.getElementById("moose-equation-7088da49-27df-4b47-adc3-832fe7e1e656");katex.render("\\vec{u}_i^\\ast", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ can be viewed as an index indicator with which the local residual goes into the global residual vector. Note that $var element = document.getElementById("moose-equation-565eeba2-7ebc-4271-bd75-1d2858941336");katex.render("\\vec{a}\\vec{b}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ represents element-wise multiplication, i.e. $var element = document.getElementById("moose-equation-1b154c2a-facd-4d6e-a2ec-b2b865c3ce75");katex.render("\\vec{a}\\vec{b}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is equal to vector whose ith element is $var element = document.getElementById("moose-equation-d557b6f2-4eda-467f-8205-c80cf647e9e4");katex.render("a_i \\times b_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , where $var element = document.getElementById("moose-equation-f4613114-87fd-4c74-9a64-eec3f86febfe");katex.render("a", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-0ff7918f-cdbc-4955-807d-5b0525b8ba69");katex.render("b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are two generic vectors.

In general, the diffusion coefficient $var element = document.getElementById("moose-equation-bf92ac44-fe3d-475d-bb74-bf25b739c1c2");katex.render("\\mathbf{D}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a 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 cooresponding to scalar, diagonal matrix and full matrix respectively. Its default value is array.

With some further transformation, the kernel becomes $var element = document.getElementById("moose-equation-d9b5ce68-a0d1-4205-8537-44793f442607");katex.render("(\\nabla \\vec{u}^\\ast, \\mathbf{D} \\nabla \\vec{u}) = \\sum_{e} \\sum_{i=1}^{N_{\\text{dof}}} \\sum_{j=1}^{N_{\\text{dof}}} \\sum_{\\text{qp}=1}^{N_{qp}} (|J|w)_{\\text{qp}} \\vec{w}_p\\vec{u}_i^\\ast \\underline{\\mathbf{D}_{\\text{qp}} \\nabla b_{j,\\text{qp}} \\cdot \\nabla b_{i,\\text{qp}}} \\vec{u}_j,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ 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.
C++ Type:MaterialPropertyName
Controllable:No
Description:The name of the diffusivity, can be scalar, vector, or matrix.
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.

Optional 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

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

Input Files

(test/tests/kernels/hfem/variable_dirichlet.i)
(test/tests/kernels/array_kernels/array_diffusion_reaction_transient.i)
(test/tests/kernels/hfem/array_dirichlet.i)
(test/tests/kernels/hfem/variable_robin.i)
(test/tests/kernels/array_kernels/array_diffusion_test.i)
(test/tests/variables/array_variable/array_variable_size_one_test.i)
(test/tests/kernels/hfem/array_dirichlet_transform.i)
(test/tests/multiapps/relaxation/picard_relaxed_array_master.i)
(test/tests/kernels/array_kernels/array_diffusion_reaction_coupling.i)
(test/tests/kernels/array_kernels/array_diffusion_reaction_other_coupling.i)
(test/tests/problems/eigen_problem/jfnk_mo/ne_array_mo.i)
(test/tests/kernels/array_kernels/array_save_in.i)
(test/tests/kernels/array_kernels/array_diffusion_reaction_dg.i)
(test/tests/auxkernels/array_var_component/array_var_component.i)
(test/tests/kernels/hfem/array_dirichlet_pjfnk.i)
(test/tests/multiapps/relaxation/picard_relaxed_array_sub.i)
(test/tests/kernels/hfem/robin_displaced.i)
(test/tests/problems/eigen_problem/arraykernels/ne_array_kernels.i)
(test/tests/transfers/multiapp_copy_transfer/array_variable_transfer/sub.i)
(test/tests/kernels/hfem/robin_adapt.i)
(test/tests/kernels/array_kernels/array_body_force.i)
(test/tests/kernels/hfem/array_dirichlet_transform_bc.i)
(test/tests/bcs/array_vacuum/array_vacuum.i)
(test/tests/kernels/hfem/array_robin.i)
(test/tests/kernels/array_kernels/array_diffusion_reaction.i)
(test/tests/problems/eigen_problem/jfnk_mo/ne_mo_with_linear_aux.i)
(test/tests/transfers/multiapp_variable_value_sample_transfer/master_array.i)
(test/tests/kernels/hfem/array_neumann.i)
(test/tests/kernels/hfem/robin_dist.i)

(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.");
  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);
}

(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.");
  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);
}

(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.");
  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);
}

(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
[]

(test/tests/kernels/hfem/variable_dirichlet.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
  [uhat]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
    uhat = uhat
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/kernels/array_kernels/array_diffusion_reaction_transient.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[Kernels]
  [dudt]
    type = ArrayTimeDerivative
    variable = u
    time_derivative_coefficient = tc
  []
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

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

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

[Materials]
  [tc]
    type = GenericConstantArray
    prop_name = tc
    prop_value = '1 1'
  []
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

[Postprocessors]
  [intu0]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 0
  []
  [intu1]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 1
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  dt = 0.1
  num_steps = 10
[]

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/array_dirichlet.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/kernels/hfem/variable_robin.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [uhat]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
  [uhat_reaction]
    type = ArrayReaction
    variable = uhat
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    reaction_coefficient = rc
  []
  [uhat_coupled]
    type = ArrayCoupledForce
    variable = uhat
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    v = lambdab
    is_v_array = true
    coef = '1 1'
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
    uhat = uhat
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [rc]
    type = GenericConstantArray
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    prop_name = rc
    prop_value = '0.5 0.5'
  []
[]

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(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
[]

(test/tests/variables/array_variable/array_variable_size_one_test.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
[]

[Variables]
  [u]
    order = FIRST
    family = MONOMIAL
    components = 1
    array = true
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
  [source]
    type = ArrayBodyForce
    variable = u
    function = '1'
  []
[]

[BCs]
  [all]
    type = ArrayVacuumBC
    boundary = 'left right top bottom'
    variable = u
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1'
  []
  [rc]
    type = GenericConstantArray
    prop_name = rc
    prop_value = '2'
  []
[]

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

[Outputs]
  [out]
    type = Exodus
  []
[]

(test/tests/kernels/hfem/array_dirichlet_transform.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusionTest
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/multiapps/relaxation/picard_relaxed_array_master.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [u]
    components = 2
  []
[]

[AuxVariables]
  [v]
    components = 2
    initial_condition = '1 1'
  []
  [inverse_v]
    components = 2
    initial_condition = '1 1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [time]
    type = ArrayTimeDerivative
    variable = u
    time_derivative_coefficient = tc
  []
  [force_u]
    type = ArrayCoupledForce
    variable = u
    v = inverse_v
    is_v_array = true
    coef = '1 1'
  []
[]

[AuxKernels]
  [invert_v]
    type = ArrayQuotientAux
    variable = inverse_v
    denominator = v
    numerator = '20 20'
  []
[]

[BCs]
  [left]
    type = ArrayDirichletBC
    variable = u
    boundary = left
    values = '0 0'
  []
  [Neumann_right]
    type = ArrayNeumannBC
    variable = u
    boundary = right
    value = '1 1'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '0.1 0.1'
  []
  [tc]
    type = GenericConstantArray
    prop_name = tc
    prop_value = '1 1'
  []
[]

[Postprocessors]
  [picard_its]
    type = NumPicardIterations
    execute_on = 'initial timestep_end'
  []
[]

[Executioner]
  type = Transient
  num_steps = 4
  dt = 0.5
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  picard_max_its = 30
  nl_abs_tol = 1e-14
  relaxation_factor = 0.8
  relaxed_variables = u
[]

[Outputs]
  exodus = true
  execute_on = 'INITIAL TIMESTEP_END'
[]

[MultiApps]
  [sub]
    type = TransientMultiApp
    execute_on = timestep_begin
    input_files = picard_relaxed_array_sub.i
  []
[]

[Transfers]
  [v_from_sub]
    type = MultiAppCopyTransfer
    from_multi_app = sub
    source_variable = v
    variable = v
  []
  [u_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = sub
    source_variable = u
    variable = u
  []
[]

(test/tests/kernels/array_kernels/array_diffusion_reaction_coupling.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
  [diffv]
    type = Diffusion
    variable = v
  []
  [vu]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '0 0.5'
  []
[]

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

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

  [leftv]
    type = DirichletBC
    variable = v
    boundary = 1
    value = 0
  []

  [rightv]
    type = DirichletBC
    variable = v
    boundary = 2
    value = 2
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

[Postprocessors]
  [intu0]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 0
  []
  [intu1]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 1
  []
  [intv]
    type = ElementIntegralVariablePostprocessor
    variable = v
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/kernels/array_kernels/array_diffusion_reaction_other_coupling.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
  [diffv]
    type = Diffusion
    variable = v
  []
  [vu]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '0 0.5'
  []
  [uv]
    type = CoupledArrayForce
    variable = v
    v = u
    coef = '0.05 0'
  []
[]

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

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

  [leftv]
    type = DirichletBC
    variable = v
    boundary = 1
    value = 0
  []

  [rightv]
    type = DirichletBC
    variable = v
    boundary = 2
    value = 2
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

[Postprocessors]
  [intu0]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 0
  []
  [intu1]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 1
  []
  [intv]
    type = ElementIntegralVariablePostprocessor
    variable = v
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/problems/eigen_problem/jfnk_mo/ne_array_mo.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

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

  [reactionu]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []

  [diffv]
    type = ArrayDiffusion
    variable = v
    diffusion_coefficient = dc
  []

  [reactionv]
    type = ArrayReaction
    variable = v
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1. 1.'
  []
  [rc]
    type = GenericConstantArray
    prop_name = rc
    prop_value = '-1 -1'
  []
[]

[BCs]
  [hom_u]
    type = ArrayDirichletBC
    variable = u
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_u]
    type = EigenArrayDirichletBC
    variable = u
    boundary = '0 1 2 3'
  []

  [hom_v]
    type = ArrayDirichletBC
    variable = v
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_v]
    type = EigenArrayDirichletBC
    variable = v
    boundary = '0 1 2 3'
  []
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNKMO
  constant_matrices = true
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
[]

(test/tests/kernels/array_kernels/array_save_in.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 4
    ny = 4
  []
  [subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '0.5 0.5 0'
    top_right = '1 1 0'
    block_id = 1
  []
[]

[Variables]
  [u]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
[]

[AuxVariables]
  [u_diff_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
  [u_vacuum_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
  [u_dg_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []

  [u_diff_diag_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
  [u_vacuum_diag_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
  [u_dg_diag_save_in]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
    save_in = u_diff_save_in
    diag_save_in = u_diff_diag_save_in
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

[DGKernels]
  [dgdiff]
    type = ArrayDGDiffusion
    variable = u
    diff = dc
    save_in = u_dg_save_in
    diag_save_in = u_dg_diag_save_in
  []
[]

[BCs]
  [left]
    type = ArrayVacuumBC
    variable = u
    boundary = 1
    save_in = u_vacuum_save_in
    diag_save_in = u_vacuum_diag_save_in
  []

  [right]
    type = ArrayPenaltyDirichletBC
    variable = u
    boundary = 2
    value = '1 2'
    penalty = 4
  []
[]

[Materials]
  [dc0]
    type = GenericConstantArray
    block = 0
    prop_name = dc
    prop_value = '1 1'
  []
  [dc1]
    type = GenericConstantArray
    block = 1
    prop_name = dc
    prop_value = '2 1'
  []
  [rc]
    type = GenericConstant2DArray
    block = '0 1'
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

[Postprocessors]
  [intu0]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 0
  []
  [intu1]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 1
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/kernels/array_kernels/array_diffusion_reaction_dg.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 4
    ny = 4
  []
  [subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '0.5 0.5 0'
    top_right = '1 1 0'
    block_id = 1
  []
[]

[Variables]
  [u]
    order = FIRST
    family = L2_LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

[DGKernels]
  [dgdiff]
    type = ArrayDGDiffusion
    variable = u
    diff = dc
  []
[]

[BCs]
  [left]
    type = ArrayVacuumBC
    variable = u
    boundary = 1
  []

  [right]
    type = ArrayPenaltyDirichletBC
    variable = u
    boundary = 2
    value = '1 2'
    penalty = 4
  []
[]

[Materials]
  [dc0]
    type = GenericConstantArray
    block = 0
    prop_name = dc
    prop_value = '1 1'
  []
  [dc1]
    type = GenericConstantArray
    block = 1
    prop_name = dc
    prop_value = '2 1'
  []
  [rc]
    type = GenericConstant2DArray
    block = '0 1'
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

[Postprocessors]
  [intu0]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 0
  []
  [intu1]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    component = 1
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/auxkernels/array_var_component/array_var_component.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[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'
  []
[]

[AuxVariables]
  [u0][]
[]

[AuxKernels]
  [u0]
    type = ArrayVariableComponent
    variable = u0
    array_variable = u
    component = 0
  []
[]

[Postprocessors]
  [intu0]
    type = ElementIntegralVariablePostprocessor
    variable = u0
  []
[]

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

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/array_dirichlet_pjfnk.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    block = 0
    reaction_coefficient = re
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusionTest
    variable = u
    lowerd_variable = lambda
    for_pjfnk = true
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletTestBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
    for_pjfnk = true
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [re]
    type = GenericConstantArray
    prop_name = re
    prop_value = '0.1 0.1'
  []
[]

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/multiapps/relaxation/picard_relaxed_array_sub.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [v]
    components = 2
  []
[]

[AuxVariables]
  [u]
    components = 2
  []
[]

[Kernels]
  [diff_v]
    type = ArrayDiffusion
    variable = v
    diffusion_coefficient = dc
  []
  [force_v]
    type = ArrayCoupledForce
    variable = v
    v = u
    is_v_array = true
    coef = '1 1'
  []
  [time_v]
    type = ArrayTimeDerivative
    variable = v
    time_derivative_coefficient = tc
  []
[]

[BCs]
  [left_v]
    type = ArrayDirichletBC
    variable = v
    boundary = left
    values = '2 2'
  []
  [right_v]
    type = ArrayDirichletBC
    variable = v
    boundary = right
    values = '1 1'
  []
[]

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

[Executioner]
  type = Transient
  num_steps = 20
  dt = 0.1
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  nl_abs_tol = 1e-10
[]

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/robin_displaced.i)

[Mesh]
  [square]
    type = CartesianMeshGenerator
    dx = '0.3125 0.3125 0.3125'
    dy = '0.3333333333333 0.3333333333333 0.3333333333333'
    dim = 2
  []
  displacements = 'x_disp y_disp'
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
  [x_disp]
    block = 0
  []
  [y_disp]
    block = 0
    initial_condition = 0
  []
[]

[AuxKernels]
  [x_disp]
    type = ParsedAux
    variable = x_disp
    use_xyzt = true
    function = 'x/15'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
    use_displaced_mesh = true
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
    use_displaced_mesh = true
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
    use_displaced_mesh = true
  []
[]

[BCs]
  [all]
    type = ArrayVacuumBC
    boundary = 'left right top bottom'
    variable = u
    use_displaced_mesh = true
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
    use_displaced_mesh = true
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    use_displaced_mesh = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/problems/eigen_problem/arraykernels/ne_array_kernels.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

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

  [reactionu]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []

  [diffv]
    type = ArrayDiffusion
    variable = v
    diffusion_coefficient = dc
  []

  [reactionv]
    type = ArrayReaction
    variable = v
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1. 1.'
  []
  [rc]
    type = GenericConstantArray
    prop_name = rc
    prop_value = '-1 -1'
  []
[]

[BCs]
  [hom_u]
    type = ArrayDirichletBC
    variable = u
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_u]
    type = EigenArrayDirichletBC
    variable = u
    boundary = '0 1 2 3'
  []

  [hom_v]
    type = ArrayDirichletBC
    variable = v
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_v]
    type = EigenArrayDirichletBC
    variable = v
    boundary = '0 1 2 3'
  []
[]

[Executioner]
  type = Eigenvalue
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
[]

(test/tests/transfers/multiapp_copy_transfer/array_variable_transfer/sub.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

[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'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/robin_adapt.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 2
    ny = 2
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayVacuumBC
    boundary = 'left right top bottom'
    variable = u
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  exodus = true
[]

[Adaptivity]
  steps = 1
  marker = box
  max_h_level = 2
  initial_steps = 2
  [Markers]
    [box]
      bottom_left = '0 0 0'
      inside = refine
      top_right = '0.5 0.5 0'
      outside = do_nothing
      type = BoxMarker
    []
  []
[]

(test/tests/kernels/array_kernels/array_body_force.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
  []
  [source]
    type = ArrayBodyForce
    variable = u
    function = '1 x'
  []
[]

[BCs]
  [left]
    type = ArrayDirichletBC
    variable = u
    boundary = 'left right bottom top'
    values = '0 0'
  []
[]

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

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

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/array_dirichlet_transform_bc.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
  [lambdab]
    order = CONSTANT
    family = MONOMIAL
    block = BOUNDARY_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayHFEMDirichletTestBC
    boundary = 'left right top bottom'
    variable = u
    lowerd_variable = lambdab
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/bcs/array_vacuum/array_vacuum.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

[BCs]
  [left]
    type = ArrayVacuumBC
    variable = u
    boundary = 1
    alpha = '1 1.2'
  []

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

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1 1'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

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

[Outputs]
  exodus = true
[]

(test/tests/kernels/hfem/array_robin.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayVacuumBC
    boundary = 'left right top bottom'
    variable = u
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/kernels/array_kernels/array_diffusion_reaction.i)

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

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
  []
[]

[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'
  []
  [rc]
    type = GenericConstant2DArray
    prop_name = rc
    prop_value = '1 0; -0.1 1'
  []
[]

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

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

[Outputs]
  exodus = true
[]

(test/tests/problems/eigen_problem/jfnk_mo/ne_mo_with_linear_aux.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[AuxVariables]
  [w][]
[]

[AuxKernels]
  [w]
    type = ArrayVariableComponent
    variable = w
    array_variable = u
    component = 0
    execute_on = linear
  []
[]

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

  [reactionu]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []

  [diffv]
    type = ArrayDiffusion
    variable = v
    diffusion_coefficient = dc
  []

  [reactionv]
    type = ArrayReaction
    variable = v
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1. 1.'
  []
  [rc]
    type = GenericConstantArray
    prop_name = rc
    prop_value = '-1 -1'
  []
[]

[BCs]
  [hom_u]
    type = ArrayDirichletBC
    variable = u
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_u]
    type = EigenArrayDirichletBC
    variable = u
    boundary = '0 1 2 3'
  []

  [hom_v]
    type = ArrayDirichletBC
    variable = v
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_v]
    type = EigenArrayDirichletBC
    variable = v
    boundary = '0 1 2 3'
  []
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNKMO
  constant_matrices = true
[]

[VectorPostprocessors]
  [eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  []
[]

[Postprocessors]
  [unorm]
    type = ElementArrayL2Norm
    variable = u
    component = 0
  []
  [wnorm]
    type = ElementL2Norm
    variable = w
  []
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
[]

(test/tests/transfers/multiapp_variable_value_sample_transfer/master_array.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
    components = 2
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayBodyForce
    variable = u
    function = '1 x'
  []
[]

[BCs]
  [left]
    type = ArrayDirichletBC
    variable = u
    boundary = 'left right bottom top'
    values = '0 0'
  []
[]

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

[Executioner]
  type = Transient
  num_steps = 1
  dt = 1

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

[MultiApps]
  [pp_sub0]
    type = TransientMultiApp
    input_files = pp_sub.i
    execute_on = timestep_end
    positions = '0.5 0.5 0 0.7 0.7 0'
  []
  [pp_sub1]
    type = TransientMultiApp
    input_files = pp_sub.i
    execute_on = timestep_end
    positions = '0.5 0.5 0 0.7 0.7 0'
  []
[]

[Transfers]
  [sample_pp_transfer0]
    type = MultiAppVariableValueSamplePostprocessorTransfer
    to_multi_app = pp_sub0
    postprocessor = from_master
    source_variable = u
    source_variable_component = 0
  []
  [sample_pp_transfer1]
    type = MultiAppVariableValueSamplePostprocessorTransfer
    to_multi_app = pp_sub1
    postprocessor = from_master
    source_variable = u
    source_variable_component = 1
  []
[]

(test/tests/kernels/hfem/array_neumann.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [reaction]
    type = ArrayReaction
    variable = u
    block = 0
    reaction_coefficient = rc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayNeumannBC
    boundary = 'left right top bottom'
    variable = u
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

(test/tests/kernels/hfem/robin_dist.i)

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    nx = 3
    ny = 3
    dim = 2
  []
  build_all_side_lowerd_mesh = true
  parallel_type = DISTRIBUTED
[]

[Variables]
  [u]
    order = THIRD
    family = MONOMIAL
    block = 0
    components = 2
  []
  [lambda]
    order = CONSTANT
    family = MONOMIAL
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
    components = 2
  []
[]

[AuxVariables]
  [v]
    order = CONSTANT
    family = MONOMIAL
    block = 0
    initial_condition = '1'
  []
[]

[Kernels]
  [diff]
    type = ArrayDiffusion
    variable = u
    block = 0
    diffusion_coefficient = dc
  []
  [source]
    type = ArrayCoupledForce
    variable = u
    v = v
    coef = '1 2'
    block = 0
  []
[]

[DGKernels]
  [surface]
    type = ArrayHFEMDiffusion
    variable = u
    lowerd_variable = lambda
  []
[]

[BCs]
  [all]
    type = ArrayVacuumBC
    boundary = 'left right top bottom'
    variable = u
  []
[]

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

[Postprocessors]
  [intu]
    type = ElementIntegralArrayVariablePostprocessor
    variable = u
    block = 0
  []
  [lambdanorm]
    type = ElementArrayL2Norm
    variable = lambda
    block = INTERNAL_SIDE_LOWERD_SUBDOMAIN
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -snes_linesearch_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu       basic                 mumps'
[]

[Outputs]
  [out]
    # we hide lambda because it may flip sign due to element
    # renumbering with distributed mesh
    type = Exodus
    hide = lambda
  []
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files

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ArrayDiffusion

Description

Example Input Syntax

Input Parameters

Required Parameters

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Input Files