matrix_tools.cc

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//
// Copyright (C) 1998 - 2021 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.
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#include <deal.II/base/function.h>
#include <deal.II/base/geometry_info.h>
#include <deal.II/base/quadrature.h>
#include <deal.II/base/work_stream.h>
 
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/dofs/dof_tools.h>
 
#include <deal.II/fe/fe.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q1.h>
 
#include <deal.II/grid/tria_iterator.h>
 
#include <deal.II/hp/fe_values.h>
#include <deal.II/hp/mapping_collection.h>
 
#include <deal.II/lac/block_sparse_matrix.h>
#include <deal.II/lac/block_vector.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/sparse_matrix.h>
#include <deal.II/lac/vector.h>
 
#include <deal.II/numerics/matrix_tools.h>
 
#ifdef DEAL_II_WITH_PETSC
#  include <deal.II/lac/petsc_block_sparse_matrix.h>
#  include <deal.II/lac/petsc_sparse_matrix.h>
#  include <deal.II/lac/petsc_vector.h>
#endif
 
#ifdef DEAL_II_WITH_TRILINOS
#  include <deal.II/lac/trilinos_block_sparse_matrix.h>
#  include <deal.II/lac/trilinos_parallel_block_vector.h>
#  include <deal.II/lac/trilinos_sparse_matrix.h>
#  include <deal.II/lac/trilinos_vector.h>
#endif
 
#include <algorithm>
#include <cmath>
#include <set>
 
 
DEAL_II_NAMESPACE_OPEN
 
 
 
namespace MatrixTools
{
  namespace
  {
    template <typename Iterator>
    bool
    column_less_than(const typename Iterator::value_type p,
                     const unsigned int                  column)
    {
      return (p.column() < column);
    }
  } // namespace
 
  // TODO:[WB] I don't think that the optimized storage of diagonals is needed
  // (GK)
  template <typename number>
  void
  apply_boundary_values(
    const std::map<types::global_dof_index, number> &boundary_values,
    SparseMatrix<number> &                           matrix,
    Vector<number> &                                 solution,
    Vector<number> &                                 right_hand_side,
    const bool                                       eliminate_columns)
  {
    Assert(matrix.n() == right_hand_side.size(),
           ExcDimensionMismatch(matrix.n(), right_hand_side.size()));
    Assert(matrix.n() == solution.size(),
           ExcDimensionMismatch(matrix.n(), solution.size()));
    Assert(matrix.n() == matrix.m(),
           ExcDimensionMismatch(matrix.n(), matrix.m()));
 
    // if no boundary values are to be applied
    // simply return
    if (boundary_values.size() == 0)
      return;
 
 
    const types::global_dof_index n_dofs = matrix.m();
 
    // if a diagonal entry is zero
    // later, then we use another
    // number instead. take it to be
    // the first nonzero diagonal
    // element of the matrix, or 1 if
    // there is no such thing
    number first_nonzero_diagonal_entry = 1;
    for (unsigned int i = 0; i < n_dofs; ++i)
      if (matrix.diag_element(i) != number())
        {
          first_nonzero_diagonal_entry = matrix.diag_element(i);
          break;
        }
 
 
    typename std::map<types::global_dof_index, number>::const_iterator
      dof  = boundary_values.begin(),
      endd = boundary_values.end();
    for (; dof != endd; ++dof)
      {
        Assert(dof->first < n_dofs, ExcInternalError());
 
        const types::global_dof_index dof_number = dof->first;
        // for each boundary dof:
 
        // set entries of this line to zero except for the diagonal
        // entry
        for (typename SparseMatrix<number>::iterator p =
               matrix.begin(dof_number);
             p != matrix.end(dof_number);
             ++p)
          if (p->column() != dof_number)
            p->value() = 0.;
 
        // set right hand side to
        // wanted value: if main diagonal
        // entry nonzero, don't touch it
        // and scale rhs accordingly. If
        // zero, take the first main
        // diagonal entry we can find, or
        // one if no nonzero main diagonal
        // element exists. Normally, however,
        // the main diagonal entry should
        // not be zero.
        //
        // store the new rhs entry to make
        // the gauss step more efficient
        number new_rhs;
        if (matrix.diag_element(dof_number) != number())
          {
            new_rhs = dof->second * matrix.diag_element(dof_number);
            right_hand_side(dof_number) = new_rhs;
          }
        else
          {
            matrix.set(dof_number, dof_number, first_nonzero_diagonal_entry);
            new_rhs = dof->second * first_nonzero_diagonal_entry;
            right_hand_side(dof_number) = new_rhs;
          }
 
 
        // if the user wants to have
        // the symmetry of the matrix
        // preserved, and if the
        // sparsity pattern is
        // symmetric, then do a Gauss
        // elimination step with the
        // present row
        if (eliminate_columns)
          {
            // store the only nonzero entry
            // of this line for the Gauss
            // elimination step
            const number diagonal_entry = matrix.diag_element(dof_number);
 
            // we have to loop over all rows of the matrix which have
            // a nonzero entry in the column which we work in
            // presently. if the sparsity pattern is symmetric, then
            // we can get the positions of these rows cheaply by
            // looking at the nonzero column numbers of the present
            // row. we need not look at the first entry of each row,
            // since that is the diagonal element and thus the present
            // row
            for (typename SparseMatrix<number>::iterator q =
                   matrix.begin(dof_number) + 1;
                 q != matrix.end(dof_number);
                 ++q)
              {
                const types::global_dof_index row = q->column();
 
                // find the position of
                // element
                // (row,dof_number)
                bool (*comp)(
                  const typename SparseMatrix<number>::iterator::value_type p,
                  const unsigned int column) =
                  &column_less_than<typename SparseMatrix<number>::iterator>;
                const typename SparseMatrix<number>::iterator p =
                  Utilities::lower_bound(matrix.begin(row) + 1,
                                         matrix.end(row),
                                         dof_number,
                                         comp);
 
                // check whether this line has an entry in the
                // regarding column (check for ==dof_number and !=
                // next_row, since if row==dof_number-1, *p is a
                // past-the-end pointer but points to dof_number
                // anyway...)
                //
                // there should be such an entry! we know this because
                // we have assumed that the sparsity pattern is
                // symmetric and we only walk over those rows for
                // which the current row has a column entry
                Assert((p != matrix.end(row)) && (p->column() == dof_number),
                       ExcMessage(
                         "This function is trying to access an element of the "
                         "matrix that doesn't seem to exist. Are you using a "
                         "nonsymmetric sparsity pattern? If so, you are not "
                         "allowed to set the eliminate_column argument of this "
                         "function, see the documentation."));
 
                // correct right hand side
                right_hand_side(row) -=
                  static_cast<number>(p->value()) / diagonal_entry * new_rhs;
 
                // set matrix entry to zero
                p->value() = 0.;
              }
          }
 
        // preset solution vector
        solution(dof_number) = dof->second;
      }
  }
 
 
 
  template <typename number>
  void
  apply_boundary_values(
    const std::map<types::global_dof_index, number> &boundary_values,
    BlockSparseMatrix<number> &                      matrix,
    BlockVector<number> &                            solution,
    BlockVector<number> &                            right_hand_side,
    const bool                                       eliminate_columns)
  {
    const unsigned int blocks = matrix.n_block_rows();
 
    Assert(matrix.n() == right_hand_side.size(),
           ExcDimensionMismatch(matrix.n(), right_hand_side.size()));
    Assert(matrix.n() == solution.size(),
           ExcDimensionMismatch(matrix.n(), solution.size()));
    Assert(matrix.n_block_rows() == matrix.n_block_cols(), ExcNotQuadratic());
    Assert(matrix.get_sparsity_pattern().get_row_indices() ==
             matrix.get_sparsity_pattern().get_column_indices(),
           ExcNotQuadratic());
    Assert(matrix.get_sparsity_pattern().get_column_indices() ==
             solution.get_block_indices(),
           ExcBlocksDontMatch());
    Assert(matrix.get_sparsity_pattern().get_row_indices() ==
             right_hand_side.get_block_indices(),
           ExcBlocksDontMatch());
 
    // if no boundary values are to be applied
    // simply return
    if (boundary_values.size() == 0)
      return;
 
 
    const types::global_dof_index n_dofs = matrix.m();
 
    // if a diagonal entry is zero
    // later, then we use another
    // number instead. take it to be
    // the first nonzero diagonal
    // element of the matrix, or 1 if
    // there is no such thing
    number first_nonzero_diagonal_entry = 0;
    for (unsigned int diag_block = 0; diag_block < blocks; ++diag_block)
      {
        for (unsigned int i = 0; i < matrix.block(diag_block, diag_block).n();
             ++i)
          if (matrix.block(diag_block, diag_block).diag_element(i) != number{})
            {
              first_nonzero_diagonal_entry =
                matrix.block(diag_block, diag_block).diag_element(i);
              break;
            }
        // check whether we have found
        // something in the present
        // block
        if (first_nonzero_diagonal_entry != number{})
          break;
      }
    // nothing found on all diagonal
    // blocks? if so, use 1.0 instead
    if (first_nonzero_diagonal_entry == number{})
      first_nonzero_diagonal_entry = 1;
 
 
    typename std::map<types::global_dof_index, number>::const_iterator
      dof  = boundary_values.begin(),
      endd = boundary_values.end();
    const BlockSparsityPattern &sparsity_pattern =
      matrix.get_sparsity_pattern();
 
    // pointer to the mapping between
    // global and block indices. since
    // the row and column mappings are
    // equal, store a pointer on only
    // one of them
    const BlockIndices &index_mapping = sparsity_pattern.get_column_indices();
 
    // now loop over all boundary dofs
    for (; dof != endd; ++dof)
      {
        Assert(dof->first < n_dofs, ExcInternalError());
        (void)n_dofs;
 
        // get global index and index
        // in the block in which this
        // dof is located
        const types::global_dof_index dof_number = dof->first;
        const std::pair<unsigned int, types::global_dof_index> block_index =
          index_mapping.global_to_local(dof_number);
 
        // for each boundary dof:
 
        // set entries of this line
        // to zero except for the diagonal
        // entry. Note that the diagonal
        // entry is always the first one
        // in a row for square matrices
        for (unsigned int block_col = 0; block_col < blocks; ++block_col)
          for (typename SparseMatrix<number>::iterator p =
                 (block_col == block_index.first ?
                    matrix.block(block_index.first, block_col)
                        .begin(block_index.second) +
                      1 :
                    matrix.block(block_index.first, block_col)
                      .begin(block_index.second));
               p != matrix.block(block_index.first, block_col)
                      .end(block_index.second);
               ++p)
            p->value() = 0;
 
        // set right hand side to
        // wanted value: if main diagonal
        // entry nonzero, don't touch it
        // and scale rhs accordingly. If
        // zero, take the first main
        // diagonal entry we can find, or
        // one if no nonzero main diagonal
        // element exists. Normally, however,
        // the main diagonal entry should
        // not be zero.
        //
        // store the new rhs entry to make
        // the gauss step more efficient
        number new_rhs;
        if (matrix.block(block_index.first, block_index.first)
              .diag_element(block_index.second) != number{})
          new_rhs =
            dof->second * matrix.block(block_index.first, block_index.first)
                            .diag_element(block_index.second);
        else
          {
            matrix.block(block_index.first, block_index.first)
              .diag_element(block_index.second) = first_nonzero_diagonal_entry;
            new_rhs = dof->second * first_nonzero_diagonal_entry;
          }
        right_hand_side.block(block_index.first)(block_index.second) = new_rhs;
 
 
        // if the user wants to have
        // the symmetry of the matrix
        // preserved, and if the
        // sparsity pattern is
        // symmetric, then do a Gauss
        // elimination step with the
        // present row. this is a
        // little more complicated for
        // block matrices.
        if (eliminate_columns)
          {
            // store the only nonzero entry
            // of this line for the Gauss
            // elimination step
            const number diagonal_entry =
              matrix.block(block_index.first, block_index.first)
                .diag_element(block_index.second);
 
            // we have to loop over all
            // rows of the matrix which
            // have a nonzero entry in
            // the column which we work
            // in presently. if the
            // sparsity pattern is
            // symmetric, then we can
            // get the positions of
            // these rows cheaply by
            // looking at the nonzero
            // column numbers of the
            // present row.
            //
            // note that if we check
            // whether row @p{row} in
            // block (r,c) is non-zero,
            // then we have to check
            // for the existence of
            // column @p{row} in block
            // (c,r), i.e. of the
            // transpose block
            for (unsigned int block_row = 0; block_row < blocks; ++block_row)
              {
                // get pointers to the sparsity patterns of this block and of
                // the transpose one
                const SparsityPattern &this_sparsity =
                  sparsity_pattern.block(block_row, block_index.first);
 
                SparseMatrix<number> &this_matrix =
                  matrix.block(block_row, block_index.first);
                SparseMatrix<number> &transpose_matrix =
                  matrix.block(block_index.first, block_row);
 
                // traverse the row of the transpose block to find the
                // interesting rows in the present block.  don't use the
                // diagonal element of the diagonal block
                for (typename SparseMatrix<number>::iterator q =
                       (block_index.first == block_row ?
                          transpose_matrix.begin(block_index.second) + 1 :
                          transpose_matrix.begin(block_index.second));
                     q != transpose_matrix.end(block_index.second);
                     ++q)
                  {
                    // get the number of the column in this row in which a
                    // nonzero entry is. this is also the row of the transpose
                    // block which has an entry in the interesting row
                    const types::global_dof_index row = q->column();
 
                    // find the position of element (row,dof_number) in this
                    // block (not in the transpose one). note that we have to
                    // take care of special cases with square sub-matrices
                    bool (*comp)(
                      typename SparseMatrix<number>::iterator::value_type p,
                      const unsigned int column) =
                      &column_less_than<
                        typename SparseMatrix<number>::iterator>;
 
                    typename SparseMatrix<number>::iterator p =
                      this_matrix.end();
 
                    if (this_sparsity.n_rows() == this_sparsity.n_cols())
                      {
                        if (this_matrix.begin(row)->column() ==
                            block_index.second)
                          p = this_matrix.begin(row);
                        else
                          p = Utilities::lower_bound(this_matrix.begin(row) + 1,
                                                     this_matrix.end(row),
                                                     block_index.second,
                                                     comp);
                      }
                    else
                      p = Utilities::lower_bound(this_matrix.begin(row),
                                                 this_matrix.end(row),
                                                 block_index.second,
                                                 comp);
 
                    // check whether this line has an entry in the
                    // regarding column (check for ==dof_number and !=
                    // next_row, since if row==dof_number-1, *p is a
                    // past-the-end pointer but points to dof_number
                    // anyway...)
                    //
                    // there should be such an entry! we know this because
                    // we have assumed that the sparsity pattern is
                    // symmetric and we only walk over those rows for
                    // which the current row has a column entry
                    Assert((p->column() == block_index.second) &&
                             (p != this_matrix.end(row)),
                           ExcInternalError());
 
                    // correct right hand side
                    right_hand_side.block(block_row)(row) -=
                      number(p->value()) / diagonal_entry * new_rhs;
 
                    // set matrix entry to zero
                    p->value() = 0.;
                  }
              }
          }
 
        // preset solution vector
        solution.block(block_index.first)(block_index.second) = dof->second;
      }
  }
 
 
 
  template <typename number>
  void
  local_apply_boundary_values(
    const std::map<types::global_dof_index, number> &boundary_values,
    const std::vector<types::global_dof_index> &     local_dof_indices,
    FullMatrix<number> &                             local_matrix,
    Vector<number> &                                 local_rhs,
    const bool                                       eliminate_columns)
  {
    Assert(local_dof_indices.size() == local_matrix.m(),
           ExcDimensionMismatch(local_dof_indices.size(), local_matrix.m()));
    Assert(local_dof_indices.size() == local_matrix.n(),
           ExcDimensionMismatch(local_dof_indices.size(), local_matrix.n()));
    Assert(local_dof_indices.size() == local_rhs.size(),
           ExcDimensionMismatch(local_dof_indices.size(), local_rhs.size()));
 
    // if there is nothing to do, then exit
    // right away
    if (boundary_values.size() == 0)
      return;
 
    // otherwise traverse all the dofs used in
    // the local matrices and vectors and see
    // what's there to do
 
    // if we need to treat an entry, then we
    // set the diagonal entry to its absolute
    // value. if it is zero, we used to set it
    // to one, which is a really terrible
    // choice that can lead to hours of
    // searching for bugs in programs (I
    // experienced this :-( ) if the matrix
    // entries are otherwise very large. this
    // is so since iterative solvers would
    // simply not correct boundary nodes for
    // their correct values since the residual
    // contributions of their rows of the
    // linear system is almost zero if the
    // diagonal entry is one. thus, set it to
    // the average absolute value of the
    // nonzero diagonal elements.
    //
    // we only compute this value lazily the
    // first time we need it.
    number             average_diagonal = 0;
    const unsigned int n_local_dofs     = local_dof_indices.size();
    for (unsigned int i = 0; i < n_local_dofs; ++i)
      {
        const typename std::map<types::global_dof_index, number>::const_iterator
          boundary_value = boundary_values.find(local_dof_indices[i]);
        if (boundary_value != boundary_values.end())
          {
            // remove this row, except for the
            // diagonal element
            for (unsigned int j = 0; j < n_local_dofs; ++j)
              if (i != j)
                local_matrix(i, j) = 0;
 
            // replace diagonal entry by its
            // absolute value to make sure that
            // everything remains positive, or
            // by the average diagonal value if
            // zero
            if (local_matrix(i, i) == number{})
              {
                // if average diagonal hasn't
                // yet been computed, do so now
                if (average_diagonal == number{})
                  {
                    unsigned int nonzero_diagonals = 0;
                    for (unsigned int k = 0; k < n_local_dofs; ++k)
                      if (local_matrix(k, k) != number{})
                        {
                          average_diagonal += std::abs(local_matrix(k, k));
                          ++nonzero_diagonals;
                        }
                    if (nonzero_diagonals != 0)
                      average_diagonal /= nonzero_diagonals;
                    else
                      average_diagonal = 0;
                  }
 
                // only if all diagonal entries
                // are zero, then resort to the
                // last measure: choose one
                if (average_diagonal == number{})
                  average_diagonal = 1.;
 
                local_matrix(i, i) = average_diagonal;
              }
            else
              local_matrix(i, i) = std::abs(local_matrix(i, i));
 
            // and replace rhs entry by correct
            // value
            local_rhs(i) = local_matrix(i, i) * boundary_value->second;
 
            // finally do the elimination step
            // if requested
            if (eliminate_columns == true)
              {
                for (unsigned int row = 0; row < n_local_dofs; ++row)
                  if (row != i)
                    {
                      local_rhs(row) -=
                        local_matrix(row, i) * boundary_value->second;
                      local_matrix(row, i) = 0;
                    }
              }
          }
      }
  }
} // namespace MatrixTools
 
 
 
// explicit instantiations
#include "matrix_tools.inst"
 
 
DEAL_II_NAMESPACE_CLOSE