// $Id$
// Version: $Name$
//
-// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 by the deal.II authors
+// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
#include <base/exceptions.h>
#include <base/quadrature_lib.h>
#include <dofs/function_map.h>
+#include <fe/mapping_q1.h>
+
#include <map>
#include <vector>
#include <set>
* following possibilities are
* implemented:
*/
- enum NormType {
+ enum NormType
+ {
/**
* The function or
* difference of functions
const Quadrature<dim-1> &q,
std::map<unsigned int,double> &boundary_values);
+
+ /**
+ * Compute the constraints that
+ * correspond to boundary conditions of
+ * the form $\vec n \cdot \vec u=0$,
+ * i.e. no normal flux if $\vec u$ is a
+ * vector-valued quantity. These
+ * conditions have exactly the form
+ * handled by the ConstraintMatrix class,
+ * so instead of creating a map between
+ * boundary degrees of freedom and
+ * corresponding value, we here create a
+ * list of constraints that are written
+ * into a ConstraintMatrix. This object
+ * may already have some content, for
+ * example from hanging node constraints,
+ * that remains untouched. These
+ * constraints have to be applied to the
+ * linear system like any other such
+ * constraints, i.e. you have to condense
+ * the linear system with the constraints
+ * before solving, and you have to
+ * distribute the solution vector
+ * afterwards.
+ *
+ * The use of this function is
+ * explained in more detail in
+ * @ref step_22 "step-22". It
+ * doesn't make much sense in 1d,
+ * so the function throws an
+ * exception in that case.
+ *
+ * The second argument of this function
+ * denotes the first vector component in
+ * the finite element that corresponds to
+ * the vector function that you want to
+ * constrain. For example, if we were
+ * solving a Stokes equation in 2d and
+ * the finite element had components
+ * $(u,v,p)$, then @p
+ * first_vector_component would be
+ * zero. On the other hand, if we solved
+ * the Maxwell equations in 3d and the
+ * finite element has components
+ * $(E_x,E_y,E_z,B_x,B_y,B_z)$ and we
+ * want the boundary condition $\vec
+ * n\cdot \vec B=0$, then @p
+ * first_vector_component would be
+ * 3. Vectors are implicitly assumed to
+ * have exactly <code>dim</code>
+ * components that are ordered in the
+ * same way as we usually order the
+ * coordinate directions, i.e. $x$-,
+ * $y$-, and finally $z$-component.
+ *
+ * The third argument denotes the set of
+ * boundary indicators on which the
+ * boundary condition is to be
+ * enforced. Note that, as explained
+ * below, this is one of the few
+ * functions where it makes a difference
+ * where we call the function multiple
+ * times with only one boundary
+ * indicator, or whether we call the
+ * function onces with the whole set of
+ * boundary indicators at once.
+ *
+ * The last argument is denoted to
+ * compute the normal vector $\vec n$ at
+ * the boundary points.
+ *
+ *
+ * <h4>Computing constraints in 2d</h4>
+ *
+ * Computing these constraints requires
+ * some smarts. The main question
+ * revolves around the question what the
+ * normal vector is. Consider the
+ * following situation:
+ * <p ALIGN="center">
+ * @image html no_normal_flux_1.png
+ * </p>
+ *
+ * Here, we have two cells that use a
+ * bilinear mapping
+ * (i.e. MappingQ1). Consequently, for
+ * each of the cells, the normal vector
+ * is perpendicular to the straight
+ * edge. If the two edges at the top and
+ * right are meant to approximate a
+ * curved boundary (as indicated by the
+ * dashed line), then neither of the two
+ * computed normal vectors are equal to
+ * the exact normal vector (though they
+ * approximate it as the mesh is refined
+ * further). What is worse, if we
+ * constrain $\vec n \cdot \vec u=0$ at
+ * the common vertex with the normal
+ * vector from both cells, then we
+ * constrain the vector $\vec u$ with
+ * respect to two linearly independent
+ * vectors; consequently, the constraint
+ * would be $\vec u=0$ at this point
+ * (i.e. <i>all</i> components of the
+ * vector), which is not what we wanted.
+ *
+ * To deal with this situation, the
+ * algorithm works in the following way:
+ * at each point where we want to
+ * constrain $\vec u$, we first collect
+ * all normal vectors that adjacent cells
+ * might compute at this point. We then
+ * do not constrain $\vec n \cdot \vec
+ * u=0$ for <i>each</i> of these normal
+ * vectors but only for the
+ * <i>average</i> of the normal
+ * vectors. In the example above, we
+ * therefore record only a single
+ * constraint $\vec n \cdot \vec {\bar
+ * u}=0$, where $\vec {\bar u}$ is the
+ * average of the two indicated normal
+ * vectors.
+ *
+ * Unfortunately, this is not quite
+ * enough. Consider the situation here:
+ *
+ * <p ALIGN="center">
+ * @image html no_normal_flux_2.png
+ * </p>
+ *
+ * If again the top and right edges
+ * approximate a curved boundary, and the
+ * left boundary a separate boundary (for
+ * example straight) so that the exact
+ * boundary has indeed a corner at the
+ * top left vertex, then the above
+ * construction would not work: here, we
+ * indeed want the constraint that $\vec
+ * u$ at this point (because the normal
+ * velocities with respect to both the
+ * left normal as well as the top normal
+ * vector should be zero), not that the
+ * velocity in the direction of the
+ * average normal vector is zero.
+ *
+ * Consequently, we use the following
+ * heuristic to determine whether all
+ * normal vectors computed at one point
+ * are to be averaged: if two normal
+ * vectors for the same point are
+ * computed on <i>different</i> cells,
+ * then they are to be averaged. This
+ * covers the first example above. If
+ * they are computed from the same cell,
+ * then the fact that they are different
+ * is considered indication that they
+ * come from different parts of the
+ * boundary that might be joined by a
+ * real corner, and must not be averaged.
+ *
+ * There is one problem with this
+ * scheme. If, for example, the same
+ * domain we have considered above, is
+ * discretized with the following mesh,
+ * then we get into trouble:
+ *
+ * <p ALIGN="center">
+ * @image html no_normal_flux_2.png
+ * </p>
+ *
+ * Here, the algorithm assumes that the
+ * boundary does not have a corner at the
+ * point where faces $F1$ and $F2$ join
+ * because at that point there are two
+ * different normal vectors computed from
+ * different cells. If you intend for
+ * there to be a corner of the exact
+ * boundary at this point, the only way
+ * to deal with this is to assign the two
+ * parts of the boundary different
+ * boundary indicators and call this
+ * function twice, once for each boundary
+ * indicators; doing so will yield only
+ * one normal vector at this point per
+ * invocation (because we consider only
+ * one boundary part at a time), with the
+ * result that the normal vectors will
+ * not be averaged.
+ *
+ *
+ * <h4>Computing constraints in 3d</h4>
+ *
+ * The situation is more
+ * complicated in 3d. Consider
+ * the following case where we
+ * want to compute the
+ * constraints at the marked
+ * vertex:
+ *
+ * <p ALIGN="center">
+ * @image html no_normal_flux_4.png
+ * </p>
+ *
+ * Here, we get four different
+ * normal vectors, one from each
+ * of the four faces that meet at
+ * the vertex. Even though they
+ * may form a complete set of
+ * vectors, it is not our intent
+ * to constrain all components of
+ * the vector field at this
+ * point. Rather, we would like
+ * to still allow tangential
+ * flow, where the term
+ * "tangential" has to be
+ * suitably defined.
+ *
+ * In a case like this, the
+ * algorithm proceeds as follows:
+ * for each cell that has
+ * computed two tangential
+ * vectors at this point, we
+ * compute the unconstrained
+ * direction as the outer product
+ * of the two tangential vectors
+ * (if necessary multiplied by
+ * minus one). We then average
+ * these tangential
+ * vectors. Finally, we compute
+ * constraints for the two
+ * directions perpendicular to
+ * this averaged tangential
+ * direction.
+ *
+ * There are cases where one cell
+ * contributes two tangential
+ * directions and another one
+ * only one; for example, this
+ * would happen if both top and
+ * front faces of the left cell
+ * belong to the boundary
+ * selected whereas only the top
+ * face of the right cell belongs
+ * to it. This case is not
+ * currently implemented.
+ */
+ template <int dim, template <int> class DH>
+ static
+ void
+ compute_no_normal_flux_constraints (const DH<dim> &dof_handler,
+ const unsigned int first_vector_component,
+ const std::set<unsigned char> &boundary_ids,
+ ConstraintMatrix &constraints,
+ const Mapping<dim> &mapping = StaticMappingQ1<dim>::mapping);
+
+
//@}
/**
* @name Assembling of right hand sides
#include <algorithm>
#include <vector>
#include <cmath>
+#include <limits>
DEAL_II_NAMESPACE_OPEN
+namespace internal
+{
+ namespace VectorTools
+ {
+ /**
+ * A structure that stores the dim DoF
+ * indices that correspond to a
+ * vector-valued quantity at a single
+ * support point.
+ */
+ template <int dim>
+ struct VectorDoFTuple
+ {
+ unsigned int dof_indices[dim];
+
+ bool operator < (const VectorDoFTuple<dim> &other) const
+ {
+ for (unsigned int i=0; i<dim; ++i)
+ if (dof_indices[i] < other.dof_indices[i])
+ return true;
+ else
+ if (dof_indices[i] > other.dof_indices[i])
+ return false;
+ return false;
+ }
+
+ bool operator == (const VectorDoFTuple<dim> &other) const
+ {
+ for (unsigned int i=0; i<dim; ++i)
+ if (dof_indices[i] != other.dof_indices[i])
+ return false;
+
+ return true;
+ }
+
+ bool operator != (const VectorDoFTuple<dim> &other) const
+ {
+ return ! (*this == other);
+ }
+ };
+
+
+
+ /**
+ * Add the constraint
+ * $\vec n \cdot \vec u = 0$
+ * to the list of constraints.
+ *
+ * Here, $\vec u$ is represented
+ * by the set of given DoF
+ * indices, and $\vec n$ by the
+ * vector specified as the second
+ * argument.
+ */
+ template <int dim>
+ void
+ add_constraint (const VectorDoFTuple<dim> &dof_indices,
+ const Tensor<1,dim> &constraining_vector,
+ ConstraintMatrix &constraints)
+ {
+ // choose the DoF that has the
+ // largest component in the
+ // constraining_vector as the
+ // one to be constrained as
+ // this makes the process
+ // stable in cases where the
+ // constraining_vector has the
+ // form n=(1,0) or n=(0,1)
+ //
+ // we get constraints of the form
+ // x0 = a_1*x1 + a2*x2 + ...
+ // if one of the weights is
+ // essentially zero then skip
+ // this part. the ConstraintMatrix
+ // can also deal with cases like
+ // x0 = 0
+ // if necessary
+ switch (dim)
+ {
+ case 2:
+ {
+ if (std::fabs(constraining_vector[0]) > std::fabs(constraining_vector[1]))
+ {
+ constraints.add_line (dof_indices.dof_indices[0]);
+
+ if (std::fabs (constraining_vector[1]/constraining_vector[0])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[0],
+ dof_indices.dof_indices[1],
+ -constraining_vector[1]/constraining_vector[0]);
+ }
+ else
+ {
+ constraints.add_line (dof_indices.dof_indices[1]);
+
+ if (std::fabs (constraining_vector[0]/constraining_vector[1])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[1],
+ dof_indices.dof_indices[0],
+ -constraining_vector[0]/constraining_vector[1]);
+ }
+ break;
+ }
+
+ case 3:
+ {
+ if ((std::fabs(constraining_vector[0]) >= std::fabs(constraining_vector[1]))
+ &&
+ (std::fabs(constraining_vector[0]) >= std::fabs(constraining_vector[2])))
+ {
+ constraints.add_line (dof_indices.dof_indices[0]);
+
+ if (std::fabs (constraining_vector[1]/constraining_vector[0])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[0],
+ dof_indices.dof_indices[1],
+ -constraining_vector[1]/constraining_vector[0]);
+
+ if (std::fabs (constraining_vector[2]/constraining_vector[0])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[0],
+ dof_indices.dof_indices[2],
+ -constraining_vector[2]/constraining_vector[0]);
+ }
+ else
+ if ((std::fabs(constraining_vector[1]) >= std::fabs(constraining_vector[0]))
+ &&
+ (std::fabs(constraining_vector[1]) >= std::fabs(constraining_vector[2])))
+ {
+ constraints.add_line (dof_indices.dof_indices[1]);
+
+ if (std::fabs (constraining_vector[0]/constraining_vector[1])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[1],
+ dof_indices.dof_indices[0],
+ -constraining_vector[0]/constraining_vector[1]);
+
+ if (std::fabs (constraining_vector[2]/constraining_vector[1])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[1],
+ dof_indices.dof_indices[2],
+ -constraining_vector[2]/constraining_vector[1]);
+ }
+ else
+ {
+ constraints.add_line (dof_indices.dof_indices[2]);
+
+ if (std::fabs (constraining_vector[0]/constraining_vector[2])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[2],
+ dof_indices.dof_indices[0],
+ -constraining_vector[0]/constraining_vector[2]);
+
+ if (std::fabs (constraining_vector[1]/constraining_vector[2])
+ > std::numeric_limits<double>::epsilon())
+ constraints.add_entry (dof_indices.dof_indices[2],
+ dof_indices.dof_indices[1],
+ -constraining_vector[1]/constraining_vector[2]);
+ }
+
+ break;
+ }
+
+ default:
+ Assert (false, ExcNotImplemented());
+ }
+
+ }
+
+
+
+ /**
+ * Given a vector, compute a set
+ * of dim-1 vectors that are
+ * orthogonal to the first one
+ * and mutually orthonormal as
+ * well.
+ */
+ template <int dim>
+ void
+ compute_orthonormal_vectors (const Tensor<1,dim> &vector,
+ Tensor<1,dim> (&orthonormals)[dim-1])
+ {
+ switch (dim)
+ {
+ case 3:
+ {
+ // to do this in 3d, take
+ // one vector that is
+ // guaranteed to be not
+ // aligned with the
+ // average tangent and
+ // form the cross
+ // product. this yields
+ // one vector that is
+ // certainly
+ // perpendicular to the
+ // tangent; then take the
+ // cross product between
+ // this vector and the
+ // tangent and get one
+ // vector that is
+ // perpendicular to both
+
+ // construct a
+ // temporary vector
+ // by swapping the
+ // larger two
+ // components and
+ // flipping one
+ // sign; this can
+ // not be collinear
+ // with the average
+ // tangent
+ Tensor<1,dim> tmp = vector;
+ if ((std::fabs(tmp[0]) > std::fabs(tmp[1]))
+ &&
+ (std::fabs(tmp[0]) > std::fabs(tmp[2])))
+ {
+ // entry zero
+ // is the
+ // largest
+ if ((std::fabs(tmp[1]) > std::fabs(tmp[2])))
+ std::swap (tmp[0], tmp[1]);
+ else
+ std::swap (tmp[0], tmp[2]);
+
+ tmp[0] *= -1;
+ }
+ else if ((std::fabs(tmp[1]) > std::fabs(tmp[0]))
+ &&
+ (std::fabs(tmp[1]) > std::fabs(tmp[2])))
+ {
+ // entry one
+ // is the
+ // largest
+ if ((std::fabs(tmp[0]) > std::fabs(tmp[2])))
+ std::swap (tmp[1], tmp[0]);
+ else
+ std::swap (tmp[1], tmp[2]);
+
+ tmp[1] *= -1;
+ }
+ else
+ {
+ // entry two
+ // is the
+ // largest
+ if ((std::fabs(tmp[0]) > std::fabs(tmp[1])))
+ std::swap (tmp[2], tmp[0]);
+ else
+ std::swap (tmp[2], tmp[1]);
+
+ tmp[2] *= -1;
+ }
+
+ Assert (std::fabs(vector * tmp) < 1e-12,
+ ExcInternalError());
+
+ // now compute the
+ // two normals
+ cross_product (orthonormals[0], vector, tmp);
+ cross_product (orthonormals[1], vector, orthonormals[0]);
+
+ break;
+ }
+
+ default:
+ Assert (false, ExcNotImplemented());
+ }
+ }
+
+ }
+}
+
+
+
+template <int dim, template <int> class DH>
+void
+VectorTools::compute_no_normal_flux_constraints (const DH<dim> &dof_handler,
+ const unsigned int first_vector_component,
+ const std::set<unsigned char> &boundary_ids,
+ ConstraintMatrix &constraints,
+ const Mapping<dim> &mapping)
+{
+ Assert (dim > 1,
+ ExcMessage ("This function is not useful in 1d because it amounts "
+ "to imposing Dirichlet values on the vector-valued "
+ "quantity."));
+
+ const FiniteElement<dim> &fe = dof_handler.get_fe();
+
+ std::vector<unsigned int> face_dofs (fe.dofs_per_face);
+ std::vector<Point<dim> > dof_locations (fe.dofs_per_face);
+
+ // have a map that stores normal vectors
+ // for each vector-dof tuple we want to
+ // constrain. since we can get at the same
+ // vector dof tuple more than once (for
+ // example if it is located at a vertex
+ // that we visit from all adjacent cells),
+ // we will want to average later on the
+ // normal vectors computed on different
+ // cells as described in the documentation
+ // of this function. however, we can only
+ // average if the contributions came from
+ // different cells, whereas we want to
+ // constrain twice or more in case the
+ // contributions came from different faces
+ // of the same cell. consequently, we also
+ // have to store which cell a normal vector
+ // was computed on
+ typedef
+ std::multimap<internal::VectorTools::VectorDoFTuple<dim>,
+ std::pair<Tensor<1,dim>, typename DH<dim>::active_cell_iterator> >
+ DoFToNormalsMap;
+
+ DoFToNormalsMap dof_to_normals_map;
+
+ // now loop over all cells and all faces
+ typename DH<dim>::active_cell_iterator
+ cell = dof_handler.begin_active(),
+ endc = dof_handler.end();
+ for (; cell!=endc; ++cell)
+ for (unsigned int face_no=0; face_no < GeometryInfo<dim>::faces_per_cell;
+ ++face_no)
+ if (boundary_ids.find(cell->face(face_no)->boundary_indicator())
+ != boundary_ids.end())
+ {
+ typename DH<dim>::face_iterator face = cell->face(face_no);
+
+ // get the indices of the
+ // dofs on this cell...
+ face->get_dof_indices (face_dofs, cell->active_fe_index());
+
+ // ...and the normal
+ // vectors at the locations
+ // where they are defined:
+ const std::vector<Point<dim-1> > &
+ unit_support_points = fe.get_unit_face_support_points();
+ Quadrature<dim-1> aux_quad (unit_support_points);
+ FEFaceValues<dim> fe_values (mapping, fe, aux_quad,
+ update_normal_vectors);
+ fe_values.reinit(cell, face_no);
+
+ // then identify which of
+ // them correspond to the
+ // selected set of vector
+ // components
+ for (unsigned int i=0; i<face_dofs.size(); ++i)
+ if (fe.face_system_to_component_index(i).first ==
+ first_vector_component)
+ {
+ // find corresponding other
+ // components of vector
+ internal::VectorTools::VectorDoFTuple<dim> vector_dofs;
+ vector_dofs.dof_indices[0] = face_dofs[i];
+
+ for (unsigned int k=0; k<fe.dofs_per_face; ++k)
+ if ((k != i)
+ &&
+ (unit_support_points[k] == unit_support_points[i])
+ &&
+ (fe.face_system_to_component_index(k).first >=
+ first_vector_component)
+ &&
+ (fe.face_system_to_component_index(k).first <
+ first_vector_component + dim))
+ vector_dofs.dof_indices[fe.face_system_to_component_index(k).first]
+ = face_dofs[k];
+
+ // and enter the
+ // (dofs,(normal_vector,cell))
+ // entry into the map
+ dof_to_normals_map
+ .insert (std::make_pair (vector_dofs,
+ std::make_pair (fe_values.normal_vector(i),
+ cell)));
+ }
+ }
+
+ // Now do something with the
+ // collected information. To this
+ // end, loop through all sets of
+ // pairs (dofs,normal_vector) and
+ // identify which entries belong to
+ // the same set of dofs and then do
+ // as described in the
+ // documentation, i.e. either
+ // average the normal vector or
+ // don't for this particular set of
+ // dofs
+ typename DoFToNormalsMap::const_iterator
+ p = dof_to_normals_map.begin();
+
+ while (p != dof_to_normals_map.end())
+ {
+ // first find the range of entries in
+ // the multimap that corresponds to the
+ // same vector-dof tuple. as usual, we
+ // define the range half-open. the
+ // first entry of course is 'p'
+ typename DoFToNormalsMap::const_iterator same_dof_range[2]
+ = { p };
+ for (++p; p != dof_to_normals_map.end(); ++p)
+ if (p->first != same_dof_range[0]->first)
+ {
+ same_dof_range[1] = p;
+ break;
+ }
+ if (p == dof_to_normals_map.end())
+ same_dof_range[1] = dof_to_normals_map.end();
+
+ // now compute the reverse mapping: for
+ // each of the cells that contributed
+ // to the current set of vector dofs,
+ // add up the normal vectors. the
+ // values of the map are pairs of
+ // normal vectors and number of cells
+ // that have contributed
+ typedef
+ std::map
+ <typename DH<dim>::active_cell_iterator,
+ std::pair<Tensor<1,dim>, unsigned int> >
+ CellToNormalsMap;
+
+ CellToNormalsMap cell_to_normals_map;
+ for (typename DoFToNormalsMap::const_iterator
+ q = same_dof_range[0];
+ q != same_dof_range[1]; ++q)
+ if (cell_to_normals_map.find (q->second.second)
+ == cell_to_normals_map.end())
+ cell_to_normals_map[q->second.second]
+ = std::make_pair (q->second.first, 1U);
+ else
+ {
+ const Tensor<1,dim> old_normal
+ = cell_to_normals_map[q->second.second].first;
+ const unsigned int old_count
+ = cell_to_normals_map[q->second.second].second;
+
+ Assert (old_count > 0, ExcInternalError());
+
+ // in the same entry,
+ // store again the now
+ // averaged normal vector
+ // and the new count
+ cell_to_normals_map[q->second.second]
+ = std::make_pair ((old_normal * old_count + q->second.first) / (old_count + 1),
+ old_count + 1);
+ }
+
+ Assert (cell_to_normals_map.size() >= 1, ExcInternalError());
+
+ // count the maximum number of
+ // contributions from each cell
+ unsigned int max_n_contributions_per_cell = 1;
+ for (typename CellToNormalsMap::const_iterator
+ x = cell_to_normals_map.begin();
+ x != cell_to_normals_map.end(); ++x)
+ max_n_contributions_per_cell
+ = std::max (max_n_contributions_per_cell,
+ x->second.second);
+
+ // verify that each cell can have only
+ // contributed at most dim times, since
+ // that is the maximum number of faces
+ // that come together at a single place
+ Assert (max_n_contributions_per_cell <= dim, ExcInternalError());
+
+ switch (max_n_contributions_per_cell)
+ {
+ // first deal with the case that a
+ // number of cells all have
+ // registered that they have a
+ // normal vector defined at the
+ // location of a given vector dof,
+ // and that each of them have
+ // encountered this vector dof
+ // exactly once while looping over
+ // all their faces. as stated in
+ // the documentation, this is the
+ // case where we want to simply
+ // average over all normal vectors
+ case 1:
+ {
+
+ // compute the average
+ // normal vector from all
+ // the ones that have the
+ // same set of dofs. we
+ // could add them up and
+ // divide them by the
+ // number of additions,
+ // or simply normalize
+ // them right away since
+ // we want them to have
+ // unit length anyway
+ Tensor<1,dim> normal;
+ for (typename CellToNormalsMap::const_iterator
+ x = cell_to_normals_map.begin();
+ x != cell_to_normals_map.end(); ++x)
+ normal += x->second.first;
+ normal /= normal.norm();
+
+ // then construct constraints
+ // from this:
+ const internal::VectorTools::VectorDoFTuple<dim> &
+ dof_indices = same_dof_range[0]->first;
+ internal::VectorTools::add_constraint (dof_indices, normal,
+ constraints);
+
+ break;
+ }
+
+
+ // this is the slightly
+ // more complicated case
+ // that a single cell has
+ // contributed with exactly
+ // DIM normal vectors to
+ // the same set of vector
+ // dofs. this is what
+ // happens in a corner in
+ // 2d and 3d (but not on an
+ // edge in 3d, where we
+ // have only 2, i.e. <DIM,
+ // contributions. Here we
+ // do not want to average
+ // the normal
+ // vectors. Since we have
+ // DIM contributions, let's
+ // assume (and verify) that
+ // they are in fact all
+ // linearly independent; in
+ // that case, all vector
+ // components are
+ // constrained and we need
+ // to set them to zero
+ case dim:
+ {
+ // assert that indeed
+ // only a single cell has
+ // contributed
+ Assert (cell_to_normals_map.size() == 1,
+ ExcInternalError());
+
+ // check linear
+ // independence by
+ // computing the
+ // determinant of the
+ // matrix created from
+ // all the normal
+ // vectors. if they are
+ // linearly independent,
+ // then the determinant
+ // is nonzero. if they
+ // are orthogonal, then
+ // the matrix is in fact
+ // equal to 1 (since they
+ // are all unit vectors);
+ // make sure the
+ // determinant is larger
+ // than 1e-3 to avoid
+ // cases where cells are
+ // degenerate
+ {
+ Tensor<2,dim> t;
+
+ typename DoFToNormalsMap::const_iterator x = same_dof_range[0];
+ for (unsigned int i=0; i<dim; ++i, ++x)
+ for (unsigned int j=0; j<dim; ++j)
+ t[i][j] = x->second.first[j];
+
+ Assert (std::fabs(determinant (t)) > 1e-3,
+ ExcMessage("Found a set of normal vectors that are nearly collinear."));
+ }
+
+ // so all components of
+ // this vector dof are
+ // constrained. enter
+ // this into the
+ // constraint matrix
+ for (unsigned int i=0; i<dim; ++i)
+ {
+ constraints.add_line (same_dof_range[0]->first.dof_indices[i]);
+ // no add_entries here
+ }
+
+ break;
+ }
+
+
+ // this is the case of an
+ // edge contribution in 3d,
+ // i.e. the vector is
+ // constrained in two
+ // directions but not the
+ // third.
+ default:
+ {
+ Assert (dim >= 3, ExcNotImplemented());
+ Assert (max_n_contributions_per_cell == 2, ExcInternalError());
+
+ // as described in the
+ // documentation, let us
+ // first collect what
+ // each of the cells
+ // contributed at the
+ // current point. we use
+ // a std::list instead of
+ // a std::set (which
+ // would be more natural)
+ // because std::set
+ // requires that the
+ // stored elements are
+ // comparable with
+ // operator<
+ typedef
+ std::map<typename DH<dim>::active_cell_iterator, std::list<Tensor<1,dim> > >
+ CellContributions;
+ CellContributions cell_contributions;
+
+ for (typename DoFToNormalsMap::const_iterator
+ q = same_dof_range[0];
+ q != same_dof_range[1]; ++q)
+ cell_contributions[q->second.second].push_back (q->second.first);
+ Assert (cell_contributions.size() >= 1, ExcInternalError());
+
+ // now for each cell that
+ // has contributed
+ // determine the number
+ // of normal vectors it
+ // has contributed. we
+ // currently only
+ // implement if this is
+ // dim-1 for all cells
+ // (if a single cell has
+ // contributed dim, or if
+ // all adjacent cells
+ // have contributed 1
+ // normal vector, this is
+ // already handled above)
+ //
+ // for each contributing
+ // cell compute the
+ // tangential vector that
+ // remains unconstrained
+ std::list<Tensor<1,dim> > tangential_vectors;
+ for (typename CellContributions::const_iterator
+ contribution = cell_contributions.begin();
+ contribution != cell_contributions.end();
+ ++contribution)
+ {
+ Assert (contribution->second.size() == dim-1, ExcNotImplemented());
+
+ Tensor<1,dim> normals[dim-1];
+ {
+ unsigned int index=0;
+ for (typename std::list<Tensor<1,dim> >::const_iterator
+ t = contribution->second.begin();
+ t != contribution->second.end();
+ ++t, ++index)
+ normals[index] = *t;
+ Assert (index == dim-1, ExcInternalError());
+ }
+
+ // calculate the
+ // tangent as the
+ // outer product of
+ // the normal vectors
+ Tensor<1,dim> tangent;
+ switch (dim)
+ {
+ case 3:
+ cross_product (tangent, normals[0], normals[1]);
+ break;
+ default:
+ Assert (false, ExcNotImplemented());
+ }
+
+ Assert (std::fabs (tangent.norm()-1) < 1e-12,
+ ExcInternalError());
+
+ tangential_vectors.push_back (tangent);
+ }
+
+ // go through the list of
+ // tangents and make sure
+ // that they all roughly
+ // point in the same
+ // direction as the first
+ // one (i.e. have an
+ // angle less than 90
+ // degrees); if they
+ // don't then flip their
+ // sign
+ {
+ const Tensor<1,dim> first_tangent = tangential_vectors.front();
+ typename std::list<Tensor<1,dim> >::iterator
+ t = tangential_vectors.begin();
+ ++t;
+ for (; t != tangential_vectors.end(); ++t)
+ if (*t * first_tangent < 0)
+ *t *= -1;
+ }
+
+ // now compute the
+ // average tangent and
+ // normalize it
+ Tensor<1,dim> average_tangent;
+ for (typename std::list<Tensor<1,dim> >::const_iterator
+ t = tangential_vectors.begin();
+ t != tangential_vectors.end();
+ ++t)
+ average_tangent += *t;
+ average_tangent /= average_tangent.norm();
+
+ // from the tangent
+ // vector we now need to
+ // again reconstruct dim-1
+ // normal directions in
+ // which the vector field
+ // is to be constrained
+ Tensor<1,dim> constraining_normals[dim-1];
+ internal::VectorTools::
+ compute_orthonormal_vectors (average_tangent,
+ constraining_normals);
+
+ // now all that is left
+ // is that we add the
+ // constraints for these
+ // dim-1 vectors
+ const internal::VectorTools::VectorDoFTuple<dim> &
+ dof_indices = same_dof_range[0]->first;
+ for (unsigned int c=0; c<dim-1; ++c)
+ internal::VectorTools::add_constraint (dof_indices,
+ constraining_normals[c],
+ constraints);
+ }
+ }
+ }
+}
+
+
+
namespace internal
{
namespace VectorTools
std::map<unsigned int,double> &);
+#if deal_II_dimension != 1
+template
+void
+VectorTools::compute_no_normal_flux_constraints (const DoFHandler<deal_II_dimension> &dof_handler,
+ const unsigned int first_vector_component,
+ const std::set<unsigned char> &boundary_ids,
+ ConstraintMatrix &constraints,
+ const Mapping<deal_II_dimension> &mapping);
+#endif
+
// // Due to introducing the DoFHandler as a template parameter,
// // the following instantiations are required in 1d
<ol>
+ <li> <p>New: The function VectorTools::compute_no_normal_flux_constraints computes
+ the constraints that correspond to boundary conditions of the
+ form $\vec u \cdot \vec n = 0$. The use of the function is demonstrated in the
+ @ref step_22 "step-22" tutorial program.
+ <br>
+ (WB 2008/01/23)
+ </p></li>
+
<li> <p>Fixed: Neither ConstraintMatrix::print nor ConstraintMatrix::write_dot
produced any output for constraints of the form $x_i=0$, i.e. where the right
hand side is a trivial linear combination of other degrees of freedom. This
############################################################
# $Id$
-# Copyright (C) 2000, 2001, 2002, 2003, 2005, 2006, 2007 by the deal.II authors
+# Copyright (C) 2000, 2001, 2002, 2003, 2005, 2006, 2007, 2008 by the deal.II authors
############################################################
############################################################
tests_x = block_matrices \
user_data_* \
constraints \
+ constraints_zero \
+ constraints_zero_merge \
+ constraints_zero_condense \
constraint_graph \
+ constraint_graph_zero \
data_out \
derivative_* \
derivatives \
maximal_cell_diameter \
union_triangulation \
create_* \
- line_coarsening_3d
+ line_coarsening_3d \
+ no_flux_*
# from above list of regular expressions, generate the real set of
# tests
--- /dev/null
+//----------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2007, 2008 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+
+
+// check the creation of no-flux boundary conditions for a finite
+// element that consists of only a single set of vector components
+// (i.e. it has dim components)
+
+#include "../tests.h"
+#include <base/logstream.h>
+#include <base/function.h>
+#include <base/quadrature_lib.h>
+#include <lac/vector.h>
+#include <grid/grid_generator.h>
+#include <dofs/dof_handler.h>
+#include <dofs/dof_constraints.h>
+#include <fe/fe_q.h>
+#include <fe/fe_system.h>
+#include <fe/mapping_q1.h>
+#include <numerics/vectors.h>
+
+#include <fstream>
+
+
+template<int dim>
+void test_projection (const Triangulation<dim>& tr,
+ const FiniteElement<dim>& fe)
+{
+ DoFHandler<dim> dof(tr);
+ dof.distribute_dofs(fe);
+
+ for (unsigned int i=0; i<GeometryInfo<dim>::faces_per_cell; ++i)
+ {
+ deallog << "FE=" << fe.get_name()
+ << ", case=" << i
+ << std::endl;
+
+ std::set<unsigned char> boundary_ids;
+ for (unsigned int j=0; j<=i; ++j)
+ boundary_ids.insert (j);
+
+ ConstraintMatrix cm;
+ VectorTools::compute_no_normal_flux_constraints (dof, 0, boundary_ids, cm);
+
+ cm.print (deallog.get_file_stream ());
+ }
+}
+
+
+template<int dim>
+void test_hyper_cube()
+{
+ Triangulation<dim> tr;
+ GridGenerator::hyper_cube(tr);
+
+ for (unsigned int i=0; i<GeometryInfo<dim>::faces_per_cell; ++i)
+ tr.begin_active()->face(i)->set_boundary_indicator (i);
+
+ tr.refine_global(2);
+
+ for (unsigned int degree=1; degree<4; ++degree)
+ {
+ FESystem<dim> fe (FE_Q<dim>(degree), dim);
+ test_projection(tr, fe);
+ }
+}
+
+
+int main()
+{
+ std::ofstream logfile ("no_flux_01/output");
+ logfile.precision (2);
+ logfile.setf(std::ios::fixed);
+ deallog.attach(logfile);
+ deallog.depth_console (0);
+ deallog.threshold_double(1.e-12);
+
+ test_hyper_cube<2>();
+ test_hyper_cube<3>();
+}
--- /dev/null
+JobId unknown Wed Jan 23 17:37:56 2008
+DEAL::FE=FESystem<2>[FE_Q<2>(1)^2], case=0
+ 0 = 0
+ 4 = 0
+ 12 = 0
+ 30 = 0
+ 36 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(1)^2], case=1
+ 0 = 0
+ 4 = 0
+ 12 = 0
+ 22 = 0
+ 24 = 0
+ 28 = 0
+ 30 = 0
+ 36 = 0
+ 44 = 0
+ 48 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(1)^2], case=2
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 9 = 0
+ 12 = 0
+ 19 = 0
+ 22 = 0
+ 23 = 0
+ 24 = 0
+ 28 = 0
+ 30 = 0
+ 36 = 0
+ 44 = 0
+ 48 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(1)^2], case=3
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 9 = 0
+ 12 = 0
+ 19 = 0
+ 22 = 0
+ 23 = 0
+ 24 = 0
+ 28 = 0
+ 30 = 0
+ 36 = 0
+ 37 = 0
+ 39 = 0
+ 41 = 0
+ 44 = 0
+ 47 = 0
+ 48 = 0
+ 49 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)^2], case=0
+ 0 = 0
+ 4 = 0
+ 8 = 0
+ 30 = 0
+ 34 = 0
+ 90 = 0
+ 94 = 0
+ 110 = 0
+ 114 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)^2], case=1
+ 0 = 0
+ 4 = 0
+ 8 = 0
+ 30 = 0
+ 34 = 0
+ 62 = 0
+ 64 = 0
+ 66 = 0
+ 82 = 0
+ 84 = 0
+ 90 = 0
+ 94 = 0
+ 110 = 0
+ 114 = 0
+ 138 = 0
+ 140 = 0
+ 154 = 0
+ 156 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)^2], case=2
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 8 = 0
+ 13 = 0
+ 19 = 0
+ 25 = 0
+ 30 = 0
+ 34 = 0
+ 51 = 0
+ 57 = 0
+ 62 = 0
+ 63 = 0
+ 64 = 0
+ 66 = 0
+ 69 = 0
+ 82 = 0
+ 84 = 0
+ 90 = 0
+ 94 = 0
+ 110 = 0
+ 114 = 0
+ 138 = 0
+ 140 = 0
+ 154 = 0
+ 156 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)^2], case=3
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 8 = 0
+ 13 = 0
+ 19 = 0
+ 25 = 0
+ 30 = 0
+ 34 = 0
+ 51 = 0
+ 57 = 0
+ 62 = 0
+ 63 = 0
+ 64 = 0
+ 66 = 0
+ 69 = 0
+ 82 = 0
+ 84 = 0
+ 90 = 0
+ 94 = 0
+ 110 = 0
+ 111 = 0
+ 113 = 0
+ 114 = 0
+ 119 = 0
+ 123 = 0
+ 127 = 0
+ 138 = 0
+ 140 = 0
+ 147 = 0
+ 151 = 0
+ 154 = 0
+ 155 = 0
+ 156 = 0
+ 159 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)^2], case=0
+ 0 = 0
+ 4 = 0
+ 8 = 0
+ 9 = 0
+ 56 = 0
+ 60 = 0
+ 61 = 0
+ 182 = 0
+ 186 = 0
+ 187 = 0
+ 224 = 0
+ 228 = 0
+ 229 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)^2], case=1
+ 0 = 0
+ 4 = 0
+ 8 = 0
+ 9 = 0
+ 56 = 0
+ 60 = 0
+ 61 = 0
+ 122 = 0
+ 124 = 0
+ 126 = 0
+ 127 = 0
+ 164 = 0
+ 166 = 0
+ 167 = 0
+ 182 = 0
+ 186 = 0
+ 187 = 0
+ 224 = 0
+ 228 = 0
+ 229 = 0
+ 284 = 0
+ 286 = 0
+ 287 = 0
+ 320 = 0
+ 322 = 0
+ 323 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)^2], case=2
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 8 = 0
+ 9 = 0
+ 18 = 0
+ 19 = 0
+ 33 = 0
+ 42 = 0
+ 43 = 0
+ 56 = 0
+ 60 = 0
+ 61 = 0
+ 99 = 0
+ 108 = 0
+ 109 = 0
+ 122 = 0
+ 123 = 0
+ 124 = 0
+ 126 = 0
+ 127 = 0
+ 132 = 0
+ 133 = 0
+ 164 = 0
+ 166 = 0
+ 167 = 0
+ 182 = 0
+ 186 = 0
+ 187 = 0
+ 224 = 0
+ 228 = 0
+ 229 = 0
+ 284 = 0
+ 286 = 0
+ 287 = 0
+ 320 = 0
+ 322 = 0
+ 323 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)^2], case=3
+ 0 = 0
+ 1 = 0
+ 3 = 0
+ 4 = 0
+ 8 = 0
+ 9 = 0
+ 18 = 0
+ 19 = 0
+ 33 = 0
+ 42 = 0
+ 43 = 0
+ 56 = 0
+ 60 = 0
+ 61 = 0
+ 99 = 0
+ 108 = 0
+ 109 = 0
+ 122 = 0
+ 123 = 0
+ 124 = 0
+ 126 = 0
+ 127 = 0
+ 132 = 0
+ 133 = 0
+ 164 = 0
+ 166 = 0
+ 167 = 0
+ 182 = 0
+ 186 = 0
+ 187 = 0
+ 224 = 0
+ 225 = 0
+ 227 = 0
+ 228 = 0
+ 229 = 0
+ 238 = 0
+ 239 = 0
+ 249 = 0
+ 256 = 0
+ 257 = 0
+ 284 = 0
+ 286 = 0
+ 287 = 0
+ 303 = 0
+ 310 = 0
+ 311 = 0
+ 320 = 0
+ 321 = 0
+ 322 = 0
+ 323 = 0
+ 328 = 0
+ 329 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=0
+ 0 = 0
+ 6 = 0
+ 12 = 0
+ 18 = 0
+ 36 = 0
+ 42 = 0
+ 54 = 0
+ 60 = 0
+ 72 = 0
+ 135 = 0
+ 141 = 0
+ 153 = 0
+ 159 = 0
+ 171 = 0
+ 180 = 0
+ 225 = 0
+ 231 = 0
+ 243 = 0
+ 252 = 0
+ 258 = 0
+ 270 = 0
+ 315 = 0
+ 324 = 0
+ 333 = 0
+ 342 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=1
+ 0 = 0
+ 6 = 0
+ 12 = 0
+ 18 = 0
+ 36 = 0
+ 42 = 0
+ 54 = 0
+ 60 = 0
+ 72 = 0
+ 93 = 0
+ 96 = 0
+ 99 = 0
+ 102 = 0
+ 111 = 0
+ 114 = 0
+ 123 = 0
+ 126 = 0
+ 132 = 0
+ 135 = 0
+ 141 = 0
+ 153 = 0
+ 159 = 0
+ 171 = 0
+ 180 = 0
+ 195 = 0
+ 198 = 0
+ 207 = 0
+ 210 = 0
+ 216 = 0
+ 222 = 0
+ 225 = 0
+ 231 = 0
+ 243 = 0
+ 252 = 0
+ 258 = 0
+ 270 = 0
+ 285 = 0
+ 288 = 0
+ 294 = 0
+ 303 = 0
+ 306 = 0
+ 312 = 0
+ 315 = 0
+ 324 = 0
+ 333 = 0
+ 342 = 0
+ 354 = 0
+ 360 = 0
+ 366 = 0
+ 372 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=2
+ 0 = 0
+ 1 = 0
+ 4 = 0
+ 6 = 0
+ 12 = 0
+ 13 = 0
+ 16 = 0
+ 18 = 0
+ 25 = 0
+ 31 = 0
+ 36 = 0
+ 42 = 0
+ 54 = 0
+ 55 = 0
+ 58 = 0
+ 60 = 0
+ 67 = 0
+ 72 = 0
+ 82 = 0
+ 88 = 0
+ 93 = 0
+ 94 = 0
+ 96 = 0
+ 99 = 0
+ 100 = 0
+ 102 = 0
+ 111 = 0
+ 114 = 0
+ 118 = 0
+ 123 = 0
+ 124 = 0
+ 126 = 0
+ 132 = 0
+ 135 = 0
+ 141 = 0
+ 153 = 0
+ 159 = 0
+ 171 = 0
+ 180 = 0
+ 195 = 0
+ 198 = 0
+ 207 = 0
+ 210 = 0
+ 216 = 0
+ 222 = 0
+ 225 = 0
+ 226 = 0
+ 229 = 0
+ 231 = 0
+ 238 = 0
+ 243 = 0
+ 252 = 0
+ 253 = 0
+ 256 = 0
+ 258 = 0
+ 265 = 0
+ 270 = 0
+ 280 = 0
+ 285 = 0
+ 286 = 0
+ 288 = 0
+ 294 = 0
+ 298 = 0
+ 303 = 0
+ 304 = 0
+ 306 = 0
+ 312 = 0
+ 315 = 0
+ 324 = 0
+ 333 = 0
+ 342 = 0
+ 354 = 0
+ 360 = 0
+ 366 = 0
+ 372 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=3
+ 0 = 0
+ 1 = 0
+ 4 = 0
+ 6 = 0
+ 12 = 0
+ 13 = 0
+ 16 = 0
+ 18 = 0
+ 25 = 0
+ 31 = 0
+ 36 = 0
+ 42 = 0
+ 54 = 0
+ 55 = 0
+ 58 = 0
+ 60 = 0
+ 67 = 0
+ 72 = 0
+ 82 = 0
+ 88 = 0
+ 93 = 0
+ 94 = 0
+ 96 = 0
+ 99 = 0
+ 100 = 0
+ 102 = 0
+ 111 = 0
+ 114 = 0
+ 118 = 0
+ 123 = 0
+ 124 = 0
+ 126 = 0
+ 132 = 0
+ 135 = 0
+ 141 = 0
+ 153 = 0
+ 154 = 0
+ 157 = 0
+ 159 = 0
+ 160 = 0
+ 163 = 0
+ 166 = 0
+ 169 = 0
+ 171 = 0
+ 180 = 0
+ 181 = 0
+ 184 = 0
+ 187 = 0
+ 195 = 0
+ 198 = 0
+ 202 = 0
+ 205 = 0
+ 207 = 0
+ 208 = 0
+ 210 = 0
+ 211 = 0
+ 216 = 0
+ 220 = 0
+ 222 = 0
+ 223 = 0
+ 225 = 0
+ 226 = 0
+ 229 = 0
+ 231 = 0
+ 238 = 0
+ 243 = 0
+ 252 = 0
+ 253 = 0
+ 256 = 0
+ 258 = 0
+ 265 = 0
+ 270 = 0
+ 280 = 0
+ 285 = 0
+ 286 = 0
+ 288 = 0
+ 294 = 0
+ 298 = 0
+ 303 = 0
+ 304 = 0
+ 306 = 0
+ 312 = 0
+ 315 = 0
+ 324 = 0
+ 325 = 0
+ 328 = 0
+ 331 = 0
+ 333 = 0
+ 342 = 0
+ 343 = 0
+ 346 = 0
+ 349 = 0
+ 354 = 0
+ 358 = 0
+ 360 = 0
+ 361 = 0
+ 366 = 0
+ 370 = 0
+ 372 = 0
+ 373 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=4
+ 0 = 0
+ 1 = 0
+ 2 = 0
+ 4 = 0
+ 5 = 0
+ 6 = 0
+ 8 = 0
+ 11 = 0
+ 12 = 0
+ 13 = 0
+ 16 = 0
+ 18 = 0
+ 25 = 0
+ 26 = 0
+ 29 = 0
+ 31 = 0
+ 36 = 0
+ 38 = 0
+ 41 = 0
+ 42 = 0
+ 50 = 0
+ 54 = 0
+ 55 = 0
+ 58 = 0
+ 60 = 0
+ 67 = 0
+ 72 = 0
+ 82 = 0
+ 83 = 0
+ 86 = 0
+ 88 = 0
+ 93 = 0
+ 94 = 0
+ 95 = 0
+ 96 = 0
+ 98 = 0
+ 99 = 0
+ 100 = 0
+ 102 = 0
+ 107 = 0
+ 111 = 0
+ 113 = 0
+ 114 = 0
+ 118 = 0
+ 123 = 0
+ 124 = 0
+ 126 = 0
+ 132 = 0
+ 135 = 0
+ 137 = 0
+ 140 = 0
+ 141 = 0
+ 149 = 0
+ 153 = 0
+ 154 = 0
+ 155 = 0
+ 157 = 0
+ 158 = 0
+ 159 = 0
+ 160 = 0
+ 163 = 0
+ 166 = 0
+ 167 = 0
+ 169 = 0
+ 171 = 0
+ 180 = 0
+ 181 = 0
+ 184 = 0
+ 187 = 0
+ 191 = 0
+ 195 = 0
+ 197 = 0
+ 198 = 0
+ 202 = 0
+ 203 = 0
+ 205 = 0
+ 207 = 0
+ 208 = 0
+ 209 = 0
+ 210 = 0
+ 211 = 0
+ 216 = 0
+ 220 = 0
+ 222 = 0
+ 223 = 0
+ 225 = 0
+ 226 = 0
+ 229 = 0
+ 231 = 0
+ 238 = 0
+ 243 = 0
+ 252 = 0
+ 253 = 0
+ 256 = 0
+ 258 = 0
+ 265 = 0
+ 270 = 0
+ 280 = 0
+ 285 = 0
+ 286 = 0
+ 288 = 0
+ 294 = 0
+ 298 = 0
+ 303 = 0
+ 304 = 0
+ 306 = 0
+ 312 = 0
+ 315 = 0
+ 324 = 0
+ 325 = 0
+ 328 = 0
+ 331 = 0
+ 333 = 0
+ 342 = 0
+ 343 = 0
+ 346 = 0
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+ 373 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(1)^3], case=5
+ 0 = 0
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=0
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=1
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=2
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=3
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=4
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)^3], case=5
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=1
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=2
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=3
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=4
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)^3], case=5
+ 0 = 0
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+ 2809 = 0
+ 2810 = 0
+ 2861 = 0
+ 2862 = 0
+ 2863 = 0
+ 2864 = 0
+ 2901 = 0
+ 2903 = 0
+ 2904 = 0
+ 2907 = 0
+ 2911 = 0
+ 2908 = 0
+ 2912 = 0
+ 2917 = 0
+ 2918 = 0
+ 2919 = 0
+ 2920 = 0
+ 2931 = 0
+ 2932 = 0
+ 2937 = 0
+ 2938 = 0
+ 2939 = 0
+ 2940 = 0
+ 2969 = 0
+ 2970 = 0
+ 2971 = 0
+ 2972 = 0
+ 3010 = 0
+ 3011 = 0
+ 3013 = 0
+ 3019 = 0
+ 3020 = 0
+ 3023 = 0
+ 3025 = 0
+ 3024 = 0
+ 3026 = 0
+ 3035 = 0
+ 3036 = 0
+ 3041 = 0
+ 3042 = 0
+ 3061 = 0
+ 3062 = 0
+ 3063 = 0
+ 3064 = 0
+ 3077 = 0
+ 3078 = 0
+ 3079 = 0
+ 3080 = 0
+ 3117 = 0
+ 3118 = 0
+ 3119 = 0
+ 3120 = 0
+ 3121 = 0
+ 3123 = 0
+ 3127 = 0
+ 3124 = 0
+ 3128 = 0
+ 3131 = 0
+ 3133 = 0
+ 3132 = 0
+ 3134 = 0
+ 3135 = 0
+ 3136 = 0
+ 3143 = 0
+ 3144 = 0
+ 3147 = 0
+ 3149 = 0
+ 3148 = 0
+ 3150 = 0
+ 3153 = 0
+ 3154 = 0
+ 3155 = 0
+ 3156 = 0
+ 3169 = 0
+ 3170 = 0
+ 3171 = 0
+ 3172 = 0
+ 3185 = 0
+ 3186 = 0
+ 3187 = 0
+ 3188 = 0
+ 3306 = 0
+ 3309 = 0
+ 3310 = 0
+ 3321 = 0
+ 3322 = 0
+ 3327 = 0
+ 3328 = 0
+ 3329 = 0
+ 3330 = 0
+ 3388 = 0
+ 3398 = 0
+ 3399 = 0
+ 3404 = 0
+ 3405 = 0
+ 3424 = 0
+ 3425 = 0
+ 3426 = 0
+ 3427 = 0
+ 3468 = 0
+ 3469 = 0
+ 3471 = 0
+ 3472 = 0
+ 3479 = 0
+ 3480 = 0
+ 3483 = 0
+ 3485 = 0
+ 3484 = 0
+ 3486 = 0
+ 3489 = 0
+ 3490 = 0
+ 3491 = 0
+ 3492 = 0
+ 3505 = 0
+ 3506 = 0
+ 3507 = 0
+ 3508 = 0
+ 3549 = 0
+ 3550 = 0
+ 3553 = 0
+ 3555 = 0
+ 3561 = 0
+ 3562 = 0
+ 3575 = 0
+ 3576 = 0
+ 3585 = 0
+ 3587 = 0
+ 3586 = 0
+ 3588 = 0
+ 3593 = 0
+ 3594 = 0
+ 3597 = 0
+ 3598 = 0
+ 3609 = 0
+ 3610 = 0
+ 3611 = 0
+ 3612 = 0
+ 3637 = 0
+ 3638 = 0
+ 3639 = 0
+ 3640 = 0
+ 3694 = 0
+ 3707 = 0
+ 3708 = 0
+ 3719 = 0
+ 3720 = 0
+ 3745 = 0
+ 3746 = 0
+ 3747 = 0
+ 3748 = 0
+ 3801 = 0
+ 3807 = 0
+ 3808 = 0
+ 3825 = 0
+ 3826 = 0
+ 3837 = 0
+ 3838 = 0
+ 3839 = 0
+ 3840 = 0
+ 3990 = 0
+ 3991 = 0
+ 3992 = 0
+ 3994 = 0
+ 3995 = 0
+ 3996 = 0
+ 3998 = 0
+ 4001 = 0
+ 4002 = 0
+ 4006 = 0
+ 4003 = 0
+ 4007 = 0
+ 4012 = 0
+ 4013 = 0
+ 4016 = 0
+ 4018 = 0
+ 4017 = 0
+ 4019 = 0
+ 4024 = 0
+ 4025 = 0
+ 4026 = 0
+ 4028 = 0
+ 4027 = 0
+ 4029 = 0
+ 4034 = 0
+ 4035 = 0
+ 4038 = 0
+ 4039 = 0
+ 4050 = 0
+ 4051 = 0
+ 4052 = 0
+ 4053 = 0
+ 4078 = 0
+ 4079 = 0
+ 4080 = 0
+ 4081 = 0
+ 4106 = 0
+ 4107 = 0
+ 4108 = 0
+ 4109 = 0
+ 4135 = 0
+ 4136 = 0
+ 4139 = 0
+ 4144 = 0
+ 4145 = 0
+ 4148 = 0
+ 4150 = 0
+ 4149 = 0
+ 4151 = 0
+ 4156 = 0
+ 4157 = 0
+ 4160 = 0
+ 4161 = 0
+ 4186 = 0
+ 4187 = 0
+ 4188 = 0
+ 4189 = 0
+ 4214 = 0
+ 4215 = 0
+ 4216 = 0
+ 4217 = 0
+ 4242 = 0
+ 4244 = 0
+ 4247 = 0
+ 4248 = 0
+ 4252 = 0
+ 4249 = 0
+ 4253 = 0
+ 4258 = 0
+ 4259 = 0
+ 4264 = 0
+ 4265 = 0
+ 4266 = 0
+ 4267 = 0
+ 4278 = 0
+ 4279 = 0
+ 4280 = 0
+ 4281 = 0
+ 4322 = 0
+ 4323 = 0
+ 4324 = 0
+ 4325 = 0
+ 4352 = 0
+ 4357 = 0
+ 4358 = 0
+ 4363 = 0
+ 4364 = 0
+ 4403 = 0
+ 4404 = 0
+ 4405 = 0
+ 4406 = 0
+ 4432 = 0
+ 4445 = 0
+ 4446 = 0
+ 4457 = 0
+ 4458 = 0
+ 4483 = 0
+ 4484 = 0
+ 4485 = 0
+ 4486 = 0
+ 4539 = 0
+ 4540 = 0
+ 4542 = 0
+ 4545 = 0
+ 4546 = 0
+ 4553 = 0
+ 4554 = 0
+ 4563 = 0
+ 4565 = 0
+ 4564 = 0
+ 4566 = 0
+ 4569 = 0
+ 4570 = 0
+ 4575 = 0
+ 4576 = 0
+ 4577 = 0
+ 4578 = 0
+ 4591 = 0
+ 4592 = 0
+ 4593 = 0
+ 4594 = 0
+ 4728 = 0
+ 4731 = 0
+ 4732 = 0
+ 4743 = 0
+ 4744 = 0
+ 4749 = 0
+ 4750 = 0
+ 4751 = 0
+ 4752 = 0
+ 4810 = 0
+ 4811 = 0
+ 4814 = 0
+ 4819 = 0
+ 4820 = 0
+ 4823 = 0
+ 4825 = 0
+ 4824 = 0
+ 4826 = 0
+ 4831 = 0
+ 4832 = 0
+ 4835 = 0
+ 4836 = 0
+ 4861 = 0
+ 4862 = 0
+ 4863 = 0
+ 4864 = 0
+ 4889 = 0
+ 4890 = 0
+ 4891 = 0
+ 4892 = 0
+ 4917 = 0
+ 4918 = 0
+ 4919 = 0
+ 4920 = 0
+ 4922 = 0
+ 4923 = 0
+ 4927 = 0
+ 4924 = 0
+ 4928 = 0
+ 4931 = 0
+ 4933 = 0
+ 4932 = 0
+ 4934 = 0
+ 4939 = 0
+ 4940 = 0
+ 4941 = 0
+ 4943 = 0
+ 4942 = 0
+ 4944 = 0
+ 4947 = 0
+ 4948 = 0
+ 4953 = 0
+ 4954 = 0
+ 4955 = 0
+ 4956 = 0
+ 4969 = 0
+ 4970 = 0
+ 4971 = 0
+ 4972 = 0
+ 4997 = 0
+ 4998 = 0
+ 4999 = 0
+ 5000 = 0
+ 5027 = 0
+ 5032 = 0
+ 5033 = 0
+ 5038 = 0
+ 5039 = 0
+ 5078 = 0
+ 5079 = 0
+ 5080 = 0
+ 5081 = 0
+ 5106 = 0
+ 5108 = 0
+ 5109 = 0
+ 5113 = 0
+ 5110 = 0
+ 5114 = 0
+ 5119 = 0
+ 5120 = 0
+ 5121 = 0
+ 5122 = 0
+ 5127 = 0
+ 5128 = 0
+ 5129 = 0
+ 5130 = 0
+ 5159 = 0
+ 5160 = 0
+ 5161 = 0
+ 5162 = 0
+ 5187 = 0
+ 5193 = 0
+ 5194 = 0
+ 5211 = 0
+ 5212 = 0
+ 5223 = 0
+ 5224 = 0
+ 5225 = 0
+ 5226 = 0
+ 5376 = 0
+ 5377 = 0
+ 5380 = 0
+ 5382 = 0
+ 5383 = 0
+ 5396 = 0
+ 5397 = 0
+ 5400 = 0
+ 5402 = 0
+ 5401 = 0
+ 5403 = 0
+ 5408 = 0
+ 5409 = 0
+ 5412 = 0
+ 5413 = 0
+ 5414 = 0
+ 5415 = 0
+ 5440 = 0
+ 5441 = 0
+ 5442 = 0
+ 5443 = 0
+ 5485 = 0
+ 5495 = 0
+ 5496 = 0
+ 5501 = 0
+ 5502 = 0
+ 5521 = 0
+ 5522 = 0
+ 5523 = 0
+ 5524 = 0
+ 5565 = 0
+ 5567 = 0
+ 5570 = 0
+ 5571 = 0
+ 5575 = 0
+ 5572 = 0
+ 5576 = 0
+ 5581 = 0
+ 5582 = 0
+ 5587 = 0
+ 5588 = 0
+ 5589 = 0
+ 5590 = 0
+ 5601 = 0
+ 5602 = 0
+ 5603 = 0
+ 5604 = 0
+ 5645 = 0
+ 5646 = 0
+ 5647 = 0
+ 5648 = 0
+ 5675 = 0
+ 5680 = 0
+ 5681 = 0
+ 5686 = 0
+ 5687 = 0
+ 5726 = 0
+ 5727 = 0
+ 5728 = 0
+ 5729 = 0
+ 5754 = 0
+ 5755 = 0
+ 5756 = 0
+ 5758 = 0
+ 5759 = 0
+ 5760 = 0
+ 5764 = 0
+ 5761 = 0
+ 5765 = 0
+ 5770 = 0
+ 5771 = 0
+ 5774 = 0
+ 5776 = 0
+ 5775 = 0
+ 5777 = 0
+ 5778 = 0
+ 5780 = 0
+ 5779 = 0
+ 5781 = 0
+ 5786 = 0
+ 5787 = 0
+ 5790 = 0
+ 5791 = 0
+ 5792 = 0
+ 5793 = 0
+ 5818 = 0
+ 5819 = 0
+ 5820 = 0
+ 5821 = 0
+ 5834 = 0
+ 5835 = 0
+ 5836 = 0
+ 5837 = 0
+ 5863 = 0
+ 5864 = 0
+ 5869 = 0
+ 5870 = 0
+ 5873 = 0
+ 5875 = 0
+ 5874 = 0
+ 5876 = 0
+ 5879 = 0
+ 5880 = 0
+ 5899 = 0
+ 5900 = 0
+ 5901 = 0
+ 5902 = 0
+ 5915 = 0
+ 5916 = 0
+ 5917 = 0
+ 5918 = 0
+ 6024 = 0
+ 6027 = 0
+ 6028 = 0
+ 6039 = 0
+ 6040 = 0
+ 6045 = 0
+ 6046 = 0
+ 6047 = 0
+ 6048 = 0
+ 6106 = 0
+ 6116 = 0
+ 6117 = 0
+ 6122 = 0
+ 6123 = 0
+ 6142 = 0
+ 6143 = 0
+ 6144 = 0
+ 6145 = 0
+ 6186 = 0
+ 6187 = 0
+ 6189 = 0
+ 6190 = 0
+ 6197 = 0
+ 6198 = 0
+ 6201 = 0
+ 6203 = 0
+ 6202 = 0
+ 6204 = 0
+ 6207 = 0
+ 6208 = 0
+ 6209 = 0
+ 6210 = 0
+ 6223 = 0
+ 6224 = 0
+ 6225 = 0
+ 6226 = 0
+ 6269 = 0
+ 6274 = 0
+ 6275 = 0
+ 6280 = 0
+ 6281 = 0
+ 6320 = 0
+ 6321 = 0
+ 6322 = 0
+ 6323 = 0
+ 6348 = 0
+ 6350 = 0
+ 6351 = 0
+ 6355 = 0
+ 6352 = 0
+ 6356 = 0
+ 6361 = 0
+ 6362 = 0
+ 6363 = 0
+ 6364 = 0
+ 6369 = 0
+ 6370 = 0
+ 6371 = 0
+ 6372 = 0
+ 6401 = 0
+ 6402 = 0
+ 6403 = 0
+ 6404 = 0
+ 6430 = 0
+ 6431 = 0
+ 6436 = 0
+ 6437 = 0
+ 6440 = 0
+ 6442 = 0
+ 6441 = 0
+ 6443 = 0
+ 6446 = 0
+ 6447 = 0
+ 6466 = 0
+ 6467 = 0
+ 6468 = 0
+ 6469 = 0
+ 6482 = 0
+ 6483 = 0
+ 6484 = 0
+ 6485 = 0
+ 6510 = 0
+ 6511 = 0
+ 6512 = 0
+ 6513 = 0
+ 6517 = 0
+ 6514 = 0
+ 6518 = 0
+ 6521 = 0
+ 6523 = 0
+ 6522 = 0
+ 6524 = 0
+ 6525 = 0
+ 6527 = 0
+ 6526 = 0
+ 6528 = 0
+ 6531 = 0
+ 6532 = 0
+ 6533 = 0
+ 6534 = 0
+ 6547 = 0
+ 6548 = 0
+ 6549 = 0
+ 6550 = 0
+ 6563 = 0
+ 6564 = 0
+ 6565 = 0
+ 6566 = 0
--- /dev/null
+//----------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2007, 2008 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+
+// check the creation of no-flux boundary conditions for a finite
+// element that consists of more than dim components and where
+// therefore we have to pick the vector components from somewhere in
+// the middle
+
+
+#include "../tests.h"
+#include <base/logstream.h>
+#include <base/function.h>
+#include <base/quadrature_lib.h>
+#include <lac/vector.h>
+#include <grid/grid_generator.h>
+#include <dofs/dof_handler.h>
+#include <dofs/dof_constraints.h>
+#include <fe/fe_q.h>
+#include <fe/fe_system.h>
+#include <fe/mapping_q1.h>
+#include <numerics/vectors.h>
+
+#include <fstream>
+
+
+
+template<int dim>
+void test_projection (const Triangulation<dim>& tr,
+ const FiniteElement<dim>& fe)
+{
+ DoFHandler<dim> dof(tr);
+ dof.distribute_dofs(fe);
+
+ for (unsigned int i=0; i<GeometryInfo<dim>::faces_per_cell; ++i)
+ {
+ deallog << "FE=" << fe.get_name()
+ << ", case=" << i
+ << std::endl;
+
+ std::set<unsigned char> boundary_ids;
+ for (unsigned int j=0; j<=i; ++j)
+ boundary_ids.insert (j);
+
+ ConstraintMatrix cm;
+ VectorTools::compute_no_normal_flux_constraints (dof, 1, boundary_ids, cm);
+
+ cm.print (deallog.get_file_stream ());
+ }
+}
+
+
+template<int dim>
+void test_hyper_cube()
+{
+ Triangulation<dim> tr;
+ GridGenerator::hyper_cube(tr);
+
+ for (unsigned int i=0; i<GeometryInfo<dim>::faces_per_cell; ++i)
+ tr.begin_active()->face(i)->set_boundary_indicator (i);
+
+ tr.refine_global(2);
+
+ for (unsigned int degree=1; degree<4; ++degree)
+ {
+ FESystem<dim> fe (FE_Q<dim>(degree+1), 1,
+ FE_Q<dim>(degree), dim,
+ FE_Q<dim>(degree+1), 1);
+ test_projection(tr, fe);
+ }
+}
+
+
+int main()
+{
+ std::ofstream logfile ("no_flux_02/output");
+ logfile.precision (2);
+ logfile.setf(std::ios::fixed);
+ deallog.attach(logfile);
+ deallog.depth_console (0);
+ deallog.threshold_double(1.e-12);
+
+ test_hyper_cube<2>();
+ test_hyper_cube<3>();
+}
--- /dev/null
+JobId unknown Wed Jan 23 17:39:41 2008
+DEAL::FE=FESystem<2>[FE_Q<2>(2)-FE_Q<2>(1)^2-FE_Q<2>(2)], case=0
+ 1 = 0
+ 9 = 0
+ 43 = 0
+ 121 = 0
+ 147 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)-FE_Q<2>(1)^2-FE_Q<2>(2)], case=1
+ 1 = 0
+ 9 = 0
+ 43 = 0
+ 85 = 0
+ 89 = 0
+ 111 = 0
+ 121 = 0
+ 147 = 0
+ 183 = 0
+ 203 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)-FE_Q<2>(1)^2-FE_Q<2>(2)], case=2
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 43 = 0
+ 85 = 0
+ 89 = 0
+ 111 = 0
+ 121 = 0
+ 147 = 0
+ 183 = 0
+ 203 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(2)-FE_Q<2>(1)^2-FE_Q<2>(2)], case=3
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 43 = 0
+ 85 = 0
+ 89 = 0
+ 111 = 0
+ 121 = 0
+ 147 = 0
+ 183 = 0
+ 203 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)-FE_Q<2>(2)^2-FE_Q<2>(3)], case=0
+ 1 = 0
+ 9 = 0
+ 18 = 0
+ 87 = 0
+ 96 = 0
+ 273 = 0
+ 282 = 0
+ 335 = 0
+ 344 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)-FE_Q<2>(2)^2-FE_Q<2>(3)], case=1
+ 1 = 0
+ 9 = 0
+ 18 = 0
+ 87 = 0
+ 96 = 0
+ 185 = 0
+ 189 = 0
+ 194 = 0
+ 247 = 0
+ 252 = 0
+ 273 = 0
+ 282 = 0
+ 335 = 0
+ 344 = 0
+ 423 = 0
+ 428 = 0
+ 475 = 0
+ 480 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)-FE_Q<2>(2)^2-FE_Q<2>(3)], case=2
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 18 = 0
+ 87 = 0
+ 96 = 0
+ 185 = 0
+ 189 = 0
+ 194 = 0
+ 247 = 0
+ 252 = 0
+ 273 = 0
+ 282 = 0
+ 335 = 0
+ 344 = 0
+ 423 = 0
+ 428 = 0
+ 475 = 0
+ 480 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(3)-FE_Q<2>(2)^2-FE_Q<2>(3)], case=3
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 18 = 0
+ 87 = 0
+ 96 = 0
+ 185 = 0
+ 189 = 0
+ 194 = 0
+ 247 = 0
+ 252 = 0
+ 273 = 0
+ 282 = 0
+ 335 = 0
+ 344 = 0
+ 423 = 0
+ 428 = 0
+ 475 = 0
+ 480 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(4)-FE_Q<2>(3)^2-FE_Q<2>(4)], case=0
+ 1 = 0
+ 9 = 0
+ 19 = 0
+ 20 = 0
+ 147 = 0
+ 157 = 0
+ 158 = 0
+ 489 = 0
+ 499 = 0
+ 500 = 0
+ 603 = 0
+ 613 = 0
+ 614 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(4)-FE_Q<2>(3)^2-FE_Q<2>(4)], case=1
+ 1 = 0
+ 9 = 0
+ 19 = 0
+ 20 = 0
+ 147 = 0
+ 157 = 0
+ 158 = 0
+ 325 = 0
+ 329 = 0
+ 335 = 0
+ 336 = 0
+ 439 = 0
+ 445 = 0
+ 446 = 0
+ 489 = 0
+ 499 = 0
+ 500 = 0
+ 603 = 0
+ 613 = 0
+ 614 = 0
+ 767 = 0
+ 773 = 0
+ 774 = 0
+ 867 = 0
+ 873 = 0
+ 874 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(4)-FE_Q<2>(3)^2-FE_Q<2>(4)], case=2
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 19 = 0
+ 20 = 0
+ 147 = 0
+ 157 = 0
+ 158 = 0
+ 325 = 0
+ 329 = 0
+ 335 = 0
+ 336 = 0
+ 439 = 0
+ 445 = 0
+ 446 = 0
+ 489 = 0
+ 499 = 0
+ 500 = 0
+ 603 = 0
+ 613 = 0
+ 614 = 0
+ 767 = 0
+ 773 = 0
+ 774 = 0
+ 867 = 0
+ 873 = 0
+ 874 = 0
+DEAL::FE=FESystem<2>[FE_Q<2>(4)-FE_Q<2>(3)^2-FE_Q<2>(4)], case=3
+ 1 = 0
+ 32767 = 0
+ 9 = 0
+ 19 = 0
+ 20 = 0
+ 147 = 0
+ 157 = 0
+ 158 = 0
+ 325 = 0
+ 329 = 0
+ 335 = 0
+ 336 = 0
+ 439 = 0
+ 445 = 0
+ 446 = 0
+ 489 = 0
+ 499 = 0
+ 500 = 0
+ 603 = 0
+ 613 = 0
+ 614 = 0
+ 767 = 0
+ 773 = 0
+ 774 = 0
+ 867 = 0
+ 873 = 0
+ 874 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=0
+ 1 = 0
+ 11 = 0
+ 21 = 0
+ 31 = 0
+ 127 = 0
+ 137 = 0
+ 205 = 0
+ 215 = 0
+ 283 = 0
+ 586 = 0
+ 596 = 0
+ 664 = 0
+ 674 = 0
+ 742 = 0
+ 791 = 0
+ 1036 = 0
+ 1046 = 0
+ 1114 = 0
+ 1163 = 0
+ 1173 = 0
+ 1241 = 0
+ 1486 = 0
+ 1535 = 0
+ 1584 = 0
+ 1633 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=1
+ 1 = 0
+ 11 = 0
+ 21 = 0
+ 31 = 0
+ 127 = 0
+ 137 = 0
+ 205 = 0
+ 215 = 0
+ 283 = 0
+ 380 = 0
+ 385 = 0
+ 390 = 0
+ 395 = 0
+ 458 = 0
+ 463 = 0
+ 518 = 0
+ 523 = 0
+ 567 = 0
+ 586 = 0
+ 596 = 0
+ 664 = 0
+ 674 = 0
+ 742 = 0
+ 791 = 0
+ 870 = 0
+ 875 = 0
+ 930 = 0
+ 935 = 0
+ 979 = 0
+ 1017 = 0
+ 1036 = 0
+ 1046 = 0
+ 1114 = 0
+ 1163 = 0
+ 1173 = 0
+ 1241 = 0
+ 1320 = 0
+ 1325 = 0
+ 1369 = 0
+ 1418 = 0
+ 1423 = 0
+ 1467 = 0
+ 1486 = 0
+ 1535 = 0
+ 1584 = 0
+ 1633 = 0
+ 1701 = 0
+ 1739 = 0
+ 1777 = 0
+ 1815 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=2
+ 1 = 0
+ 32767 = 0
+ 11 = 0
+ 21 = 0
+ 31 = 0
+ 127 = 0
+ 137 = 0
+ 205 = 0
+ 215 = 0
+ 283 = 0
+ 380 = 0
+ 385 = 0
+ 390 = 0
+ 395 = 0
+ 458 = 0
+ 463 = 0
+ 518 = 0
+ 523 = 0
+ 567 = 0
+ 586 = 0
+ 596 = 0
+ 664 = 0
+ 674 = 0
+ 742 = 0
+ 791 = 0
+ 870 = 0
+ 875 = 0
+ 930 = 0
+ 935 = 0
+ 979 = 0
+ 1017 = 0
+ 1036 = 0
+ 1046 = 0
+ 1114 = 0
+ 1163 = 0
+ 1173 = 0
+ 1241 = 0
+ 1320 = 0
+ 1325 = 0
+ 1369 = 0
+ 1418 = 0
+ 1423 = 0
+ 1467 = 0
+ 1486 = 0
+ 1535 = 0
+ 1584 = 0
+ 1633 = 0
+ 1701 = 0
+ 1739 = 0
+ 1777 = 0
+ 1815 = 0
+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=3
+ 1 = 0
+ 32767 = 0
+ 11 = 0
+ 21 = 0
+ 31 = 0
+ 127 = 0
+ 137 = 0
+ 205 = 0
+ 215 = 0
+ 283 = 0
+ 380 = 0
+ 385 = 0
+ 390 = 0
+ 395 = 0
+ 458 = 0
+ 463 = 0
+ 518 = 0
+ 523 = 0
+ 567 = 0
+ 586 = 0
+ 596 = 0
+ 664 = 0
+ 674 = 0
+ 742 = 0
+ 791 = 0
+ 870 = 0
+ 875 = 0
+ 930 = 0
+ 935 = 0
+ 979 = 0
+ 1017 = 0
+ 1036 = 0
+ 1046 = 0
+ 1114 = 0
+ 1163 = 0
+ 1173 = 0
+ 1241 = 0
+ 1320 = 0
+ 1325 = 0
+ 1369 = 0
+ 1418 = 0
+ 1423 = 0
+ 1467 = 0
+ 1486 = 0
+ 1535 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=4
+ 1 = 0
+ 32767 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(2)-FE_Q<3>(1)^3-FE_Q<3>(2)], case=5
+ 1 = 0
+ 32767 = 0
+ 2 = 0
+ 7 = 0
+ 11 = 0
+ 12 = 0
+ 17 = 0
+ 21 = 0
+ 31 = 0
+ 80 = 0
+ 85 = 0
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+ 137 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=0
+ 1 = 0
+ 11 = 0
+ 21 = 0
+ 31 = 0
+ 42 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=1
+ 1 = 0
+ 11 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=2
+ 1 = 0
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=3
+ 1 = 0
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=4
+ 1 = 0
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(3)-FE_Q<3>(2)^3-FE_Q<3>(3)], case=5
+ 1 = 0
+ 0 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=0
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=1
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=2
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=3
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=4
+ 1 = 0
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+DEAL::FE=FESystem<3>[FE_Q<3>(4)-FE_Q<3>(3)^3-FE_Q<3>(4)], case=5
+ 1 = 0
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