--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 1998 - 2020 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+#ifndef dealii_matrix_creator_h
+#define dealii_matrix_creator_h
+
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/exceptions.h>
+#include <deal.II/base/function.h>
+
+#include <deal.II/lac/affine_constraints.h>
+
+#include <map>
+
+#ifdef DEAL_II_WITH_PETSC
+# include <petscsys.h>
+#endif
+
+DEAL_II_NAMESPACE_OPEN
+
+
+// forward declarations
+#ifndef DOXYGEN
+template <int dim>
+class Quadrature;
+
+
+template <typename number>
+class Vector;
+template <typename number>
+class SparseMatrix;
+
+template <int dim, int spacedim>
+class Mapping;
+template <int dim, int spacedim>
+class DoFHandler;
+
+namespace hp
+{
+ template <int>
+ class QCollection;
+ template <int, int>
+ class MappingCollection;
+} // namespace hp
+
+#endif
+
+
+/**
+ * This namespace provides functions that assemble certain standard matrices
+ * for a given triangulation, using a given finite element, a given mapping
+ * and a quadrature formula.
+ *
+ *
+ * <h3>Conventions for all functions</h3>
+ *
+ * There exist two versions of almost all functions, one that takes an
+ * explicit Mapping argument and one that does not. The second one generally
+ * calls the first with an implicit $Q_1$ argument (i.e., with an argument of
+ * kind MappingQ(1)). If your intend your code to use a different
+ * mapping than a (bi-/tri-)linear one, then you need to call the functions
+ * <b>with</b> mapping argument should be used.
+ *
+ * All functions take a sparse matrix object to hold the matrix to be created.
+ * The functions assume that the matrix is initialized with a sparsity pattern
+ * (SparsityPattern) corresponding to the given degree of freedom handler,
+ * i.e. the sparsity structure is already as needed. You can do this by
+ * calling the DoFTools::make_sparsity_pattern() function.
+ *
+ * Furthermore it is assumed that no relevant data is in the matrix. Some
+ * entries will be overwritten and some others will contain invalid data if
+ * the matrix wasn't empty before. Therefore you may want to clear the matrix
+ * before assemblage.
+ *
+ * By default, all created matrices are `raw': they are not condensed, i.e.
+ * hanging nodes are not eliminated. The reason is that you may want to add
+ * several matrices and could then condense afterwards only once, instead of
+ * for every matrix. To actually do computations with these matrices, you have
+ * to condense the matrix using the AffineConstraints::condense function; you
+ * also have to condense the right hand side accordingly and distribute the
+ * solution afterwards. Alternatively, you can give an optional argument
+ * AffineConstraints that writes cell matrix (and vector) entries with
+ * distribute_local_to_global into the global matrix and vector. This way,
+ * adding several matrices from different sources is more complicated and
+ * you should make sure that you do not mix different ways of applying
+ * constraints. Particular caution is necessary when the given
+ * AffineConstraints object contains inhomogeneous constraints: In that case,
+ * the matrix assembled this way must be the only matrix (or you need to
+ * assemble the <b>same</b> right hand side for <b>every</b> matrix you
+ * generate and add together).
+ *
+ * If you want to use boundary conditions with the matrices generated by the
+ * functions of this namespace in addition to the ones in a possible
+ * AffineConstraints object, you have to use a function like
+ * <tt>apply_boundary_values</tt> with the matrix, solution, and right hand
+ * side.
+ *
+ *
+ * <h3>Supported matrices</h3>
+ *
+ * At present there are functions to create the following matrices:
+ * <ul>
+ * <li> @p create_mass_matrix: create the matrix with entries $m_{ij} =
+ * \int_\Omega \phi_i(x) \phi_j(x) dx$ by numerical quadrature. Here, the
+ * $\phi_i$ are the basis functions of the finite element space given.
+ *
+ * A coefficient may be given to evaluate $m_{ij} = \int_\Omega a(x) \phi_i(x)
+ * \phi_j(x) dx$ instead.
+ *
+ * <li> @p create_laplace_matrix: create the matrix with entries $a_{ij} =
+ * \int_\Omega \nabla\phi_i(x) \nabla\phi_j(x) dx$ by numerical quadrature.
+ *
+ * Again, a coefficient may be given to evaluate $a_{ij} = \int_\Omega a(x)
+ * \nabla\phi_i(x) \nabla\phi_j(x) dx$ instead.
+ * </ul>
+ *
+ * Make sure that the order of the Quadrature formula given to these functions
+ * is sufficiently high to compute the matrices with the required accuracy.
+ * For the choice of this quadrature rule you need to take into account the
+ * polynomial degree of the FiniteElement basis functions, the roughness of
+ * the coefficient @p a, as well as the degree of the given @p Mapping (if
+ * any).
+ *
+ * Note, that for vector-valued elements the mass matrix and the laplace
+ * matrix is implemented in such a way that each component couples only with
+ * itself, i.e. there is no coupling of shape functions belonging to different
+ * components. If the degrees of freedom have been sorted according to their
+ * vector component (e.g., using DoFRenumbering::component_wise()), then the
+ * resulting matrices will be block diagonal.
+ *
+ * If the finite element for which the mass matrix or the Laplace matrix is to
+ * be built has more than one component, the functions accept a single
+ * coefficient as well as a vector valued coefficient function. For the latter
+ * case, the number of components must coincide with the number of components
+ * of the system finite element.
+ *
+ *
+ * <h3>Matrices on the boundary</h3>
+ *
+ * The create_boundary_mass_matrix() creates the matrix with entries $m_{ij} =
+ * \int_{\Gamma} \phi_i \phi_j dx$, where $\Gamma$ is the union of boundary
+ * parts with indicators contained in a std::map<types::boundary_id, const
+ * Function<spacedim,number>*> passed to the function (i.e. if you want to set
+ * up the mass matrix for the parts of the boundary with indicators zero and 2,
+ * you pass the function a map with key type types::boundary_id
+ * as the parameter @p boundary_functions containing the keys zero and
+ * 2). The size of the matrix is equal to the number of degrees of freedom
+ * that have support on the boundary, i.e. it is <em>not</em> a matrix on all
+ * degrees of freedom, but only a subset. (The $\phi_i$ in the formula are the
+ * subset of basis functions which have at least part of their support on
+ * $\Gamma$.) In order to determine which shape functions are to be
+ * considered, and in order to determine in which order, the function takes a
+ * @p dof_to_boundary_mapping; this object maps global DoF numbers to a
+ * numbering of the degrees of freedom located on the boundary, and can be
+ * obtained using the function DoFTools::map_dof_to_boundary_indices().
+ *
+ * In order to work, the function needs a matrix of the correct size, built on
+ * top of a corresponding sparsity pattern. Since we only work on a subset of
+ * the degrees of freedom, we can't use the matrices and sparsity patterns
+ * that are created for the entire set of degrees of freedom. Rather, you
+ * should use the DoFHandler::make_boundary_sparsity_pattern() function to
+ * create the correct sparsity pattern, and build a matrix on top of it.
+ *
+ * Note that at present there is no function that computes the mass matrix for
+ * <em>all</em> shape functions, though such a function would be trivial to
+ * implement.
+ *
+ *
+ * <h3>Right hand sides</h3>
+ *
+ * In many cases, you will not only want to build the matrix, but also a right
+ * hand side, which will give a vector with $f_i = \int_\Omega f(x) \phi_i(x)
+ * dx$. For this purpose, each function exists in two versions, one only
+ * building the matrix and one also building the right hand side vector. If
+ * you want to create a right hand side vector without creating a matrix, you
+ * can use the VectorTools::create_right_hand_side() function. The use of the
+ * latter may be useful if you want to create many right hand side vectors.
+ *
+ * @ingroup numerics
+ */
+namespace MatrixCreator
+{
+ /**
+ * Assemble the mass matrix. If no coefficient is given (i.e., if the
+ * pointer to a function object is zero as it is by default), the
+ * coefficient is taken as being constant and equal to one.
+ * In case you want to specify @p constraints and use the default argument
+ * for the coefficient you have to specify the (unused) coefficient argument
+ * as <code>(const Function<spacedim,number> *const)nullptr</code>.
+ *
+ * If the library is configured to use multithreading, this function works
+ * in parallel.
+ *
+ * The optional argument @p constraints allows to apply constraints on the
+ * resulting matrix directly. Note, however, that this becomes difficult
+ * when you have inhomogeneous constraints and later want to add several
+ * such matrices, for example in time dependent settings such as the main
+ * loop of step-26.
+ *
+ * See the general documentation of this namespace for more information.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Call the create_mass_matrix() function, see above, with
+ * <tt>mapping=MappingQ@<dim@>(1)</tt>.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Assemble the mass matrix and a right hand side vector. If no coefficient
+ * is given (i.e., if the pointer to a function object is zero as it is by
+ * default), the coefficient is taken as being constant and equal to one.
+ * In case you want to specify @p constraints and use the default argument
+ * for the coefficient you have to specify the (unused) coefficient argument
+ * as <code>(const Function <spacedim,number> *const)nullptr</code>.
+ *
+ * If the library is configured to use multithreading, this function works
+ * in parallel.
+ *
+ * The optional argument @p constraints allows to apply constraints on the
+ * resulting matrix directly. Note, however, that this becomes difficult
+ * when you have inhomogeneous constraints and later want to add several
+ * such matrices, for example in time dependent settings such as the main
+ * loop of step-26.
+ *
+ * See the general documentation of this namespace for more information.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Call the create_mass_matrix() function, see above, with
+ * <tt>mapping=MappingQ@<dim@>(1)</tt>.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const DoFHandler<dim, spacedim> & dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const hp::MappingCollection<dim, spacedim> &mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const hp::MappingCollection<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_mass_matrix(
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+
+ /**
+ * Assemble the mass matrix and a right hand side vector along the boundary.
+ *
+ * The matrix is assumed to already be initialized with a suiting sparsity
+ * pattern (the DoFHandler provides an appropriate function).
+ *
+ * If the library is configured to use multithreading, this function works
+ * in parallel.
+ *
+ * @arg @p weight: an optional weight for the computation of the mass
+ * matrix. If no weight is given, it is set to one.
+ * In case you want to specify @p component_mapping and use the default argument
+ * for the coefficient you have to specify the (unused) coefficient argument
+ * as <code>(const Function <spacedim,number> *const)nullptr</code>.
+ *
+ * @arg @p component_mapping: if the components in @p boundary_functions and
+ * @p dof do not coincide, this vector allows them to be remapped. If the
+ * vector is not empty, it has to have one entry for each component in @p
+ * dof. This entry is the component number in @p boundary_functions that
+ * should be used for this component in @p dof. By default, no remapping is
+ * applied.
+ *
+ * @todo This function does not work for finite elements with cell-dependent
+ * shape functions.
+ */
+ template <int dim, int spacedim, typename number>
+ void
+ create_boundary_mass_matrix(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim - 1> & q,
+ SparseMatrix<number> & matrix,
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ & boundary_functions,
+ Vector<number> & rhs_vector,
+ std::vector<types::global_dof_index> & dof_to_boundary_mapping,
+ const Function<spacedim, number> *const weight = 0,
+ std::vector<unsigned int> component_mapping = {});
+
+
+ /**
+ * Call the create_boundary_mass_matrix() function, see above, with
+ * <tt>mapping=MappingQ@<dim@>(1)</tt>.
+ */
+ template <int dim, int spacedim, typename number>
+ void
+ create_boundary_mass_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim - 1> & q,
+ SparseMatrix<number> & matrix,
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ & boundary_functions,
+ Vector<number> & rhs_vector,
+ std::vector<types::global_dof_index> & dof_to_boundary_mapping,
+ const Function<spacedim, number> *const a = nullptr,
+ std::vector<unsigned int> component_mapping = {});
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename number>
+ void
+ create_boundary_mass_matrix(
+ const hp::MappingCollection<dim, spacedim> &mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim - 1> & q,
+ SparseMatrix<number> & matrix,
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ & boundary_functions,
+ Vector<number> & rhs_vector,
+ std::vector<types::global_dof_index> & dof_to_boundary_mapping,
+ const Function<spacedim, number> *const a = nullptr,
+ std::vector<unsigned int> component_mapping = {});
+
+ /**
+ * Same function as above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename number>
+ void
+ create_boundary_mass_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const hp::QCollection<dim - 1> & q,
+ SparseMatrix<number> & matrix,
+ const std::map<types::boundary_id, const Function<spacedim, number> *>
+ & boundary_functions,
+ Vector<number> & rhs_vector,
+ std::vector<types::global_dof_index> & dof_to_boundary_mapping,
+ const Function<spacedim, number> *const a = nullptr,
+ std::vector<unsigned int> component_mapping = {});
+
+ /**
+ * Assemble the Laplace matrix. If no coefficient is given (i.e., if the
+ * pointer to a function object is zero as it is by default), the
+ * coefficient is taken as being constant and equal to one.
+ * In case you want to specify @p constraints and use the default argument
+ * for the coefficient you have to specify the (unused) coefficient argument
+ * as <code>(const Function<spacedim> *const)nullptr</code>.
+ *
+ * If the library is configured to use multithreading, this function works
+ * in parallel.
+ *
+ * The optional argument @p constraints allows to apply constraints on the
+ * resulting matrix directly. Note, however, that this becomes difficult
+ * when you have inhomogeneous constraints and later want to add several
+ * such matrices, for example in time dependent settings such as the main
+ * loop of step-26.
+ *
+ * See the general documentation of this namespace for more information.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Call the create_laplace_matrix() function, see above, with
+ * <tt>mapping=MappingQ@<dim@>(1)</tt>.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Assemble the Laplace matrix and a right hand side vector. If no
+ * coefficient is given, it is assumed to be constant one.
+ * In case you want to specify @p constraints and use the default argument
+ * for the coefficient you have to specify the (unused) coefficient argument
+ * as <code>(const Function<spacedim> *const)nullptr</code>.
+ *
+ * If the library is configured to use multithreading, this function works
+ * in parallel.
+ *
+ * The optional argument @p constraints allows to apply constraints on the
+ * resulting matrix directly. Note, however, that this becomes difficult
+ * when you have inhomogeneous constraints and later want to add several
+ * such matrices, for example in time dependent settings such as the main
+ * loop of step-26.
+ *
+ * See the general documentation of this namespace for more information.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const Mapping<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Call the create_laplace_matrix() function, see above, with
+ * <tt>mapping=MappingQ@<dim@>(1)</tt>.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const DoFHandler<dim, spacedim> & dof,
+ const Quadrature<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Like the functions above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const hp::MappingCollection<dim, spacedim> &mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Like the functions above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const DoFHandler<dim, spacedim> &dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Like the functions above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const hp::MappingCollection<dim, spacedim> & mapping,
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Like the functions above, but for hp-objects.
+ */
+ template <int dim, int spacedim, typename SparseMatrixType>
+ void
+ create_laplace_matrix(
+ const DoFHandler<dim, spacedim> & dof,
+ const hp::QCollection<dim> & q,
+ SparseMatrixType & matrix,
+ const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
+ Vector<typename SparseMatrixType::value_type> & rhs_vector,
+ const Function<spacedim, typename SparseMatrixType::value_type> *const a =
+ nullptr,
+ const AffineConstraints<typename SparseMatrixType::value_type> &
+ constraints = AffineConstraints<typename SparseMatrixType::value_type>());
+
+ /**
+ * Exception
+ */
+ DeclExceptionMsg(ExcComponentMismatch,
+ "You are providing either a right hand side function or a "
+ "coefficient with a number of vector components that is "
+ "inconsistent with the rest of the arguments. If you do "
+ "provide a coefficient or right hand side function, then "
+ "it either needs to have as many components as the finite "
+ "element in use, or only a single vector component. In "
+ "the latter case, the same value will be taken for "
+ "each vector component of the finite element.");
+} // namespace MatrixCreator
+
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/numerics/matrix_creator.h>
+
#include <map>
#ifdef DEAL_II_WITH_PETSC
#endif
-/**
- * This namespace provides functions that assemble certain standard matrices
- * for a given triangulation, using a given finite element, a given mapping
- * and a quadrature formula.
- *
- *
- * <h3>Conventions for all functions</h3>
- *
- * There exist two versions of almost all functions, one that takes an
- * explicit Mapping argument and one that does not. The second one generally
- * calls the first with an implicit $Q_1$ argument (i.e., with an argument of
- * kind MappingQ(1)). If your intend your code to use a different
- * mapping than a (bi-/tri-)linear one, then you need to call the functions
- * <b>with</b> mapping argument should be used.
- *
- * All functions take a sparse matrix object to hold the matrix to be created.
- * The functions assume that the matrix is initialized with a sparsity pattern
- * (SparsityPattern) corresponding to the given degree of freedom handler,
- * i.e. the sparsity structure is already as needed. You can do this by
- * calling the DoFTools::make_sparsity_pattern() function.
- *
- * Furthermore it is assumed that no relevant data is in the matrix. Some
- * entries will be overwritten and some others will contain invalid data if
- * the matrix wasn't empty before. Therefore you may want to clear the matrix
- * before assemblage.
- *
- * By default, all created matrices are `raw': they are not condensed, i.e.
- * hanging nodes are not eliminated. The reason is that you may want to add
- * several matrices and could then condense afterwards only once, instead of
- * for every matrix. To actually do computations with these matrices, you have
- * to condense the matrix using the AffineConstraints::condense function; you
- * also have to condense the right hand side accordingly and distribute the
- * solution afterwards. Alternatively, you can give an optional argument
- * AffineConstraints that writes cell matrix (and vector) entries with
- * distribute_local_to_global into the global matrix and vector. This way,
- * adding several matrices from different sources is more complicated and
- * you should make sure that you do not mix different ways of applying
- * constraints. Particular caution is necessary when the given
- * AffineConstraints object contains inhomogeneous constraints: In that case,
- * the matrix assembled this way must be the only matrix (or you need to
- * assemble the <b>same</b> right hand side for <b>every</b> matrix you
- * generate and add together).
- *
- * If you want to use boundary conditions with the matrices generated by the
- * functions of this namespace in addition to the ones in a possible
- * AffineConstraints object, you have to use a function like
- * <tt>apply_boundary_values</tt> with the matrix, solution, and right hand
- * side.
- *
- *
- * <h3>Supported matrices</h3>
- *
- * At present there are functions to create the following matrices:
- * <ul>
- * <li> @p create_mass_matrix: create the matrix with entries $m_{ij} =
- * \int_\Omega \phi_i(x) \phi_j(x) dx$ by numerical quadrature. Here, the
- * $\phi_i$ are the basis functions of the finite element space given.
- *
- * A coefficient may be given to evaluate $m_{ij} = \int_\Omega a(x) \phi_i(x)
- * \phi_j(x) dx$ instead.
- *
- * <li> @p create_laplace_matrix: create the matrix with entries $a_{ij} =
- * \int_\Omega \nabla\phi_i(x) \nabla\phi_j(x) dx$ by numerical quadrature.
- *
- * Again, a coefficient may be given to evaluate $a_{ij} = \int_\Omega a(x)
- * \nabla\phi_i(x) \nabla\phi_j(x) dx$ instead.
- * </ul>
- *
- * Make sure that the order of the Quadrature formula given to these functions
- * is sufficiently high to compute the matrices with the required accuracy.
- * For the choice of this quadrature rule you need to take into account the
- * polynomial degree of the FiniteElement basis functions, the roughness of
- * the coefficient @p a, as well as the degree of the given @p Mapping (if
- * any).
- *
- * Note, that for vector-valued elements the mass matrix and the laplace
- * matrix is implemented in such a way that each component couples only with
- * itself, i.e. there is no coupling of shape functions belonging to different
- * components. If the degrees of freedom have been sorted according to their
- * vector component (e.g., using DoFRenumbering::component_wise()), then the
- * resulting matrices will be block diagonal.
- *
- * If the finite element for which the mass matrix or the Laplace matrix is to
- * be built has more than one component, the functions accept a single
- * coefficient as well as a vector valued coefficient function. For the latter
- * case, the number of components must coincide with the number of components
- * of the system finite element.
- *
- *
- * <h3>Matrices on the boundary</h3>
- *
- * The create_boundary_mass_matrix() creates the matrix with entries $m_{ij} =
- * \int_{\Gamma} \phi_i \phi_j dx$, where $\Gamma$ is the union of boundary
- * parts with indicators contained in a std::map<types::boundary_id, const
- * Function<spacedim,number>*> passed to the function (i.e. if you want to set
- * up the mass matrix for the parts of the boundary with indicators zero and 2,
- * you pass the function a map with key type types::boundary_id
- * as the parameter @p boundary_functions containing the keys zero and
- * 2). The size of the matrix is equal to the number of degrees of freedom
- * that have support on the boundary, i.e. it is <em>not</em> a matrix on all
- * degrees of freedom, but only a subset. (The $\phi_i$ in the formula are the
- * subset of basis functions which have at least part of their support on
- * $\Gamma$.) In order to determine which shape functions are to be
- * considered, and in order to determine in which order, the function takes a
- * @p dof_to_boundary_mapping; this object maps global DoF numbers to a
- * numbering of the degrees of freedom located on the boundary, and can be
- * obtained using the function DoFTools::map_dof_to_boundary_indices().
- *
- * In order to work, the function needs a matrix of the correct size, built on
- * top of a corresponding sparsity pattern. Since we only work on a subset of
- * the degrees of freedom, we can't use the matrices and sparsity patterns
- * that are created for the entire set of degrees of freedom. Rather, you
- * should use the DoFHandler::make_boundary_sparsity_pattern() function to
- * create the correct sparsity pattern, and build a matrix on top of it.
- *
- * Note that at present there is no function that computes the mass matrix for
- * <em>all</em> shape functions, though such a function would be trivial to
- * implement.
- *
- *
- * <h3>Right hand sides</h3>
- *
- * In many cases, you will not only want to build the matrix, but also a right
- * hand side, which will give a vector with $f_i = \int_\Omega f(x) \phi_i(x)
- * dx$. For this purpose, each function exists in two versions, one only
- * building the matrix and one also building the right hand side vector. If
- * you want to create a right hand side vector without creating a matrix, you
- * can use the VectorTools::create_right_hand_side() function. The use of the
- * latter may be useful if you want to create many right hand side vectors.
- *
- * @ingroup numerics
- */
-namespace MatrixCreator
-{
- /**
- * Assemble the mass matrix. If no coefficient is given (i.e., if the
- * pointer to a function object is zero as it is by default), the
- * coefficient is taken as being constant and equal to one.
- * In case you want to specify @p constraints and use the default argument
- * for the coefficient you have to specify the (unused) coefficient argument
- * as <code>(const Function<spacedim,number> *const)nullptr</code>.
- *
- * If the library is configured to use multithreading, this function works
- * in parallel.
- *
- * The optional argument @p constraints allows to apply constraints on the
- * resulting matrix directly. Note, however, that this becomes difficult
- * when you have inhomogeneous constraints and later want to add several
- * such matrices, for example in time dependent settings such as the main
- * loop of step-26.
- *
- * See the general documentation of this namespace for more information.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const Mapping<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Call the create_mass_matrix() function, see above, with
- * <tt>mapping=MappingQ@<dim@>(1)</tt>.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Assemble the mass matrix and a right hand side vector. If no coefficient
- * is given (i.e., if the pointer to a function object is zero as it is by
- * default), the coefficient is taken as being constant and equal to one.
- * In case you want to specify @p constraints and use the default argument
- * for the coefficient you have to specify the (unused) coefficient argument
- * as <code>(const Function <spacedim,number> *const)nullptr</code>.
- *
- * If the library is configured to use multithreading, this function works
- * in parallel.
- *
- * The optional argument @p constraints allows to apply constraints on the
- * resulting matrix directly. Note, however, that this becomes difficult
- * when you have inhomogeneous constraints and later want to add several
- * such matrices, for example in time dependent settings such as the main
- * loop of step-26.
- *
- * See the general documentation of this namespace for more information.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const Mapping<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> & dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Call the create_mass_matrix() function, see above, with
- * <tt>mapping=MappingQ@<dim@>(1)</tt>.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const DoFHandler<dim, spacedim> & dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const hp::MappingCollection<dim, spacedim> &mapping,
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const hp::MappingCollection<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_mass_matrix(
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
-
- /**
- * Assemble the mass matrix and a right hand side vector along the boundary.
- *
- * The matrix is assumed to already be initialized with a suiting sparsity
- * pattern (the DoFHandler provides an appropriate function).
- *
- * If the library is configured to use multithreading, this function works
- * in parallel.
- *
- * @arg @p weight: an optional weight for the computation of the mass
- * matrix. If no weight is given, it is set to one.
- * In case you want to specify @p component_mapping and use the default argument
- * for the coefficient you have to specify the (unused) coefficient argument
- * as <code>(const Function <spacedim,number> *const)nullptr</code>.
- *
- * @arg @p component_mapping: if the components in @p boundary_functions and
- * @p dof do not coincide, this vector allows them to be remapped. If the
- * vector is not empty, it has to have one entry for each component in @p
- * dof. This entry is the component number in @p boundary_functions that
- * should be used for this component in @p dof. By default, no remapping is
- * applied.
- *
- * @todo This function does not work for finite elements with cell-dependent
- * shape functions.
- */
- template <int dim, int spacedim, typename number>
- void
- create_boundary_mass_matrix(
- const Mapping<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim - 1> & q,
- SparseMatrix<number> & matrix,
- const std::map<types::boundary_id, const Function<spacedim, number> *>
- & boundary_functions,
- Vector<number> & rhs_vector,
- std::vector<types::global_dof_index> & dof_to_boundary_mapping,
- const Function<spacedim, number> *const weight = 0,
- std::vector<unsigned int> component_mapping = {});
-
-
- /**
- * Call the create_boundary_mass_matrix() function, see above, with
- * <tt>mapping=MappingQ@<dim@>(1)</tt>.
- */
- template <int dim, int spacedim, typename number>
- void
- create_boundary_mass_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim - 1> & q,
- SparseMatrix<number> & matrix,
- const std::map<types::boundary_id, const Function<spacedim, number> *>
- & boundary_functions,
- Vector<number> & rhs_vector,
- std::vector<types::global_dof_index> & dof_to_boundary_mapping,
- const Function<spacedim, number> *const a = nullptr,
- std::vector<unsigned int> component_mapping = {});
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename number>
- void
- create_boundary_mass_matrix(
- const hp::MappingCollection<dim, spacedim> &mapping,
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim - 1> & q,
- SparseMatrix<number> & matrix,
- const std::map<types::boundary_id, const Function<spacedim, number> *>
- & boundary_functions,
- Vector<number> & rhs_vector,
- std::vector<types::global_dof_index> & dof_to_boundary_mapping,
- const Function<spacedim, number> *const a = nullptr,
- std::vector<unsigned int> component_mapping = {});
-
- /**
- * Same function as above, but for hp-objects.
- */
- template <int dim, int spacedim, typename number>
- void
- create_boundary_mass_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const hp::QCollection<dim - 1> & q,
- SparseMatrix<number> & matrix,
- const std::map<types::boundary_id, const Function<spacedim, number> *>
- & boundary_functions,
- Vector<number> & rhs_vector,
- std::vector<types::global_dof_index> & dof_to_boundary_mapping,
- const Function<spacedim, number> *const a = nullptr,
- std::vector<unsigned int> component_mapping = {});
-
- /**
- * Assemble the Laplace matrix. If no coefficient is given (i.e., if the
- * pointer to a function object is zero as it is by default), the
- * coefficient is taken as being constant and equal to one.
- * In case you want to specify @p constraints and use the default argument
- * for the coefficient you have to specify the (unused) coefficient argument
- * as <code>(const Function<spacedim> *const)nullptr</code>.
- *
- * If the library is configured to use multithreading, this function works
- * in parallel.
- *
- * The optional argument @p constraints allows to apply constraints on the
- * resulting matrix directly. Note, however, that this becomes difficult
- * when you have inhomogeneous constraints and later want to add several
- * such matrices, for example in time dependent settings such as the main
- * loop of step-26.
- *
- * See the general documentation of this namespace for more information.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const Mapping<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Call the create_laplace_matrix() function, see above, with
- * <tt>mapping=MappingQ@<dim@>(1)</tt>.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Assemble the Laplace matrix and a right hand side vector. If no
- * coefficient is given, it is assumed to be constant one.
- * In case you want to specify @p constraints and use the default argument
- * for the coefficient you have to specify the (unused) coefficient argument
- * as <code>(const Function<spacedim> *const)nullptr</code>.
- *
- * If the library is configured to use multithreading, this function works
- * in parallel.
- *
- * The optional argument @p constraints allows to apply constraints on the
- * resulting matrix directly. Note, however, that this becomes difficult
- * when you have inhomogeneous constraints and later want to add several
- * such matrices, for example in time dependent settings such as the main
- * loop of step-26.
- *
- * See the general documentation of this namespace for more information.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const Mapping<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> & dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Call the create_laplace_matrix() function, see above, with
- * <tt>mapping=MappingQ@<dim@>(1)</tt>.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const DoFHandler<dim, spacedim> & dof,
- const Quadrature<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Like the functions above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const hp::MappingCollection<dim, spacedim> &mapping,
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Like the functions above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const DoFHandler<dim, spacedim> &dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Like the functions above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const hp::MappingCollection<dim, spacedim> & mapping,
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Like the functions above, but for hp-objects.
- */
- template <int dim, int spacedim, typename SparseMatrixType>
- void
- create_laplace_matrix(
- const DoFHandler<dim, spacedim> & dof,
- const hp::QCollection<dim> & q,
- SparseMatrixType & matrix,
- const Function<spacedim, typename SparseMatrixType::value_type> &rhs,
- Vector<typename SparseMatrixType::value_type> & rhs_vector,
- const Function<spacedim, typename SparseMatrixType::value_type> *const a =
- nullptr,
- const AffineConstraints<typename SparseMatrixType::value_type> &
- constraints = AffineConstraints<typename SparseMatrixType::value_type>());
-
- /**
- * Exception
- */
- DeclExceptionMsg(ExcComponentMismatch,
- "You are providing either a right hand side function or a "
- "coefficient with a number of vector components that is "
- "inconsistent with the rest of the arguments. If you do "
- "provide a coefficient or right hand side function, then "
- "it either needs to have as many components as the finite "
- "element in use, or only a single vector component. In "
- "the latter case, the same value will be taken for "
- "each vector component of the finite element.");
-} // namespace MatrixCreator
-
-
/**
* Provide a collection of functions operating on matrices. These include the
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