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// $Id: mapping_q1.h 20158 2009-11-24 00:03:32Z kronbichler $
// Version: $Name$
//
// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 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.
//
//---------------------------------------------------------------------------
#ifndef __deal2__mapping_q1_h
#define __deal2__mapping_q1_h
#include <base/config.h>
#include <base/table.h>
#include <base/qprojector.h>
#include <grid/tria_iterator.h>
#include <dofs/dof_accessor.h>
#include <fe/mapping.h>
#include <cmath>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup mapping */
/*@{*/
/**
* Mapping of general quadrilateral/hexahedra by d-linear shape
* functions.
*
* This function maps the unit cell to a general grid cell with
* straight lines in $d$ dimensions (remark that in 3D the surfaces
* may be curved, even if the edges are not). This is the well-known
* mapping for polyhedral domains.
*
* Shape function for this mapping are the same as for the finite
* element FE_Q of order 1. Therefore, coupling these two yields
* an isoparametric element.
*
* For more information about the <tt>spacedim</tt> template parameter
* check the documentation of FiniteElement or the one of
* Triangulation.
*
* @author Guido Kanschat, 2000, 2001; Ralf Hartmann, 2000, 2001, 2005
*/
template <int dim, int spacedim=dim>
class MappingQ1 : public Mapping<dim,spacedim>
{
public:
/**
* Default constructor.
*/
MappingQ1 ();
virtual Point<spacedim>
transform_unit_to_real_cell (
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<dim> &p) const;
/**
* Transforms the point @p p on
* the real cell to the point
* @p p_unit on the unit cell
* @p cell and returns @p p_unit.
*
* Uses Newton iteration and the
* @p transform_unit_to_real_cell
* function.
*/
virtual Point<dim>
transform_real_to_unit_cell (
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<spacedim> &p) const;
virtual void
transform (const VectorSlice<const std::vector<Tensor<1,dim> > > input,
VectorSlice<std::vector<Tensor<1,spacedim> > > output,
const typename Mapping<dim,spacedim>::InternalDataBase &internal,
const MappingType type) const;
virtual void
transform (const VectorSlice<const std::vector<Tensor<2,dim> > > input,
VectorSlice<std::vector<Tensor<2,spacedim> > > output,
const typename Mapping<dim,spacedim>::InternalDataBase &internal,
const MappingType type) const;
/**
* Return a pointer to a copy of the
* present object. The caller of this
* copy then assumes ownership of it.
*/
virtual
Mapping<dim,spacedim> * clone () const;
/**
* Storage for internal data of
* d-linear transformation.
*/
class InternalData : public Mapping<dim,spacedim>::InternalDataBase
{
public:
/**
* Constructor. Pass the
* number of shape functions.
*/
InternalData(const unsigned int n_shape_functions);
/**
* Shape function at quadrature
* point. Shape functions are
* in tensor product order, so
* vertices must be reordered
* to obtain transformation.
*/
double shape (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Shape function at quadrature
* point. See above.
*/
double &shape (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Gradient of shape function
* in quadrature point. See
* above.
*/
Tensor<1,dim> derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Gradient of shape function
* in quadrature point. See
* above.
*/
Tensor<1,dim> &derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Second derivative of shape
* function in quadrature
* point. See above.
*/
Tensor<2,dim> second_derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Second derivative of shape
* function in quadrature
* point. See above.
*/
Tensor<2,dim> &second_derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Return an estimate (in
* bytes) or the memory
* consumption of this
* object.
*/
virtual unsigned int memory_consumption () const;
/**
* Values of shape
* functions. Access by
* function @p shape.
*
* Computed once.
*/
std::vector<double> shape_values;
/**
* Values of shape function
* derivatives. Access by
* function @p derivative.
*
* Computed once.
*/
std::vector<Tensor<1,dim> > shape_derivatives;
/**
* Values of shape function
* second derivatives. Access
* by function
* @p second_derivative.
*
* Computed once.
*/
std::vector<Tensor<2,dim> > shape_second_derivatives;
/**
* Tensors of covariant
* transformation at each of
* the quadrature points. The
* matrix stored is the
* inverse of the Jacobian
* matrix, which itself is
* stored in the
* @p contravariant field of
* this structure.
*
* Computed on each cell.
*/
std::vector<Tensor<2,spacedim> > covariant;
/**
* Tensors of contravariant
* transformation at each of
* the quadrature points. The
* contravariant matrix is
* the Jacobian of the
* transformation,
* i.e. $J_{ij}=dx_i/d\hat x_j$.
*
* Computed on each cell.
*/
std::vector<Tensor<2,spacedim> > contravariant;
/**
* Unit tangential vectors. Used
* for the computation of
* boundary forms and normal
* vectors.
*
* Filled once.
*/
std::vector<std::vector<Tensor<1,spacedim> > > unit_tangentials;
/**
* Auxiliary vectors for internal use.
*/
std::vector<std::vector<Tensor<1,spacedim> > > aux;
/**
* Stores the support points of
* the mapping shape functions on
* the @p cell_of_current_support_points.
*/
std::vector<Point<spacedim> > mapping_support_points;
/**
* Stores the cell of which the
* @p mapping_support_points are
* stored.
*/
typename Triangulation<dim,spacedim>::cell_iterator cell_of_current_support_points;
/**
* Default value of this flag
* is @p true. If <tt>*this</tt>
* is an object of a derived
* class, this flag is set to
* @p false.
*/
bool is_mapping_q1_data;
/**
* Number of shape
* functions. If this is a Q1
* mapping, then it is simply
* the number of vertices per
* cell. However, since also
* derived classes use this
* class (e.g. the
* Mapping_Q() class),
* the number of shape
* functions may also be
* different.
*/
unsigned int n_shape_functions;
};
/**
* Declare a convenience typedef
* for the class that describes
* offsets into quadrature
* formulas projected onto faces
* and subfaces.
*/
typedef
typename QProjector<dim>::DataSetDescriptor
DataSetDescriptor;
/**
* Implementation of the interface in
* Mapping.
*/
virtual void
fill_fe_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Quadrature<dim> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
typename std::vector<Point<spacedim> > &quadrature_points,
std::vector<double> &JxW_values,
std::vector<Tensor<2,spacedim> > &jacobians,
std::vector<Tensor<3,spacedim> > &jacobian_grads,
std::vector<Tensor<2,spacedim> > &inverse_jacobians,
std::vector<Point<spacedim> > &cell_normal_vectors,
CellSimilarity::Similarity &cell_similarity) const;
/**
* Implementation of the interface in
* Mapping.
*/
virtual void
fill_fe_face_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const Quadrature<dim-1> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
typename std::vector<Point<dim> > &quadrature_points,
std::vector<double> &JxW_values,
typename std::vector<Tensor<1,dim> > &boundary_form,
typename std::vector<Point<spacedim> > &normal_vectors) const ;
/**
* Implementation of the interface in
* Mapping.
*/
virtual void
fill_fe_subface_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int sub_no,
const Quadrature<dim-1>& quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
typename std::vector<Point<dim> > &quadrature_points,
std::vector<double> &JxW_values,
typename std::vector<Tensor<1,dim> > &boundary_form,
typename std::vector<Point<spacedim> > &normal_vectors) const ;
/**
* Compute shape values and/or
* derivatives.
*
* Calls either the
* @p compute_shapes_virtual of
* this class or that of the
* derived class, depending on
* whether
* <tt>data.is_mapping_q1_data</tt>
* equals @p true or @p false.
*/
void compute_shapes (const std::vector<Point<dim> > &unit_points,
InternalData &data) const;
/**
* Do the computations for the
* @p get_data functions. Here,
* the data vectors of
* @p InternalData are
* reinitialized to proper size
* and shape values are computed.
*/
void compute_data (const UpdateFlags flags,
const Quadrature<dim> &quadrature,
const unsigned int n_orig_q_points,
InternalData &data) const;
/**
* Do the computations for the
* @p get_face_data
* functions. Here, the data
* vectors of @p InternalData
* are reinitialized to proper
* size and shape values and
* derivatives are
* computed. Furthermore
* @p unit_tangential vectors of
* the face are computed.
*/
void compute_face_data (const UpdateFlags flags,
const Quadrature<dim> &quadrature,
const unsigned int n_orig_q_points,
InternalData &data) const;
/**
* Do the computation for the
* <tt>fill_*</tt> functions.
*/
void compute_fill (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int npts,
const DataSetDescriptor data_set,
const CellSimilarity::Similarity cell_similarity,
InternalData &data,
std::vector<Point<spacedim> > &quadrature_points) const;
/**
* Do the computation for the
* <tt>fill_*</tt> functions.
*/
void compute_fill_face (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int subface_no,
const unsigned int npts,
const DataSetDescriptor data_set,
const std::vector<double> &weights,
InternalData &mapping_data,
std::vector<Point<dim> > &quadrature_points,
std::vector<double> &JxW_values,
std::vector<Tensor<1,dim> > &boundary_form,
std::vector<Point<spacedim> > &normal_vectors) const;
/**
* Compute shape values and/or
* derivatives.
*/
virtual void compute_shapes_virtual (const std::vector<Point<dim> > &unit_points,
InternalData &data) const;
/**
* Transforms a point @p p on
* the unit cell to the point
* @p p_real on the real cell
* @p cell and returns @p p_real.
*
* This function is called by
* @p transform_unit_to_real_cell
* and multiply (through the
* Newton iteration) by
* @p transform_real_to_unit_cell_internal.
*
* Takes a reference to an
* @p InternalData that must
* already include the shape
* values at point @p p and the
* mapping support points of the
* cell.
*
* This @p InternalData argument
* avoids multiple computations
* of the shape values at point
* @p p and especially multiple
* computations of the mapping
* support points.
*/
Point<spacedim> transform_unit_to_real_cell_internal (const InternalData &mdata) const;
/**
* Transforms the point @p p on
* the real cell to the point
* @p p_unit on the unit cell
* @p cell by a Newton
* iteration.
*
* Takes a reference to an
* @p InternalData that is
* assumed to be previously
* created by the @p get_data
* function with @p UpdateFlags
* including
* @p update_transformation_values
* and
* @p update_transformation_gradients
* and a one point Quadrature
* including the given point
* @p p_unit. Hence this
* function assumes that
* @p mdata already includes the
* transformation shape values
* and gradients computed at
* @p p_unit.
*
* These assumptions should be
* fulfilled by the calling
* function. That is up to now
* only the function
* @p transform_real_to_unit_cell
* and its overloaded versions.
* @p mdata will be changed by
* this function.
*/
void transform_real_to_unit_cell_internal (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<spacedim> &p,
InternalData &mdata,
Point<dim> &p_unit) const;
/**
* Always returns @p true because
* MappingQ1 preserves vertex locations.
*/
virtual
bool preserves_vertex_locations () const;
private:
/**
* Implementation of the interface in
* Mapping.
*
* Description of effects:
* <ul>
* <li> if @p update_quadrature_points
* is required, the output will
* contain
* @p update_transformation_values. This
* computes the values of the
* transformation basis
* polynomials at the unit cell
* quadrature points.
* <li> if any of
* @p update_covariant_transformation,
* @p update_contravariant_transformation,
* @p update_JxW_values,
* @p update_boundary_forms,
* @p update_normal_vectors is
* required, the output will
* contain
* @p update_transformation_gradients
* to compute derivatives of the
* transformation basis
* polynomials.
* </ul>
*/
virtual UpdateFlags update_once (const UpdateFlags flags) const;
/**
* Implementation of the interface in
* Mapping.
*
* Description of effects if
* @p flags contains:
* <ul>
* <li> <code>update_quadrature_points</code> is
* copied to the output to
* compute the quadrature points
* on the real cell.
* <li> <code>update_JxW_values</code> is
* copied and requires
* @p update_boundary_forms on
* faces. The latter, because the
* surface element is just the
* norm of the boundary form.
* <li> <code>update_normal_vectors</code>
* is copied and requires
* @p update_boundary_forms. The
* latter, because the normal
* vector is the normalized
* boundary form.
* <li>
* <code>update_covariant_transformation</code>
* is copied and requires
* @p update_contravariant_transformation,
* since it is computed as the
* inverse of the latter.
* <li> <code>update_JxW_values</code> is
* copied and requires
* <code>update_contravariant_transformation</code>,
* since it is computed as one
* over determinant of the
* latter.
* <li> <code>update_boundary_forms</code>
* is copied and requires
* <code>update_contravariant_transformation</code>,
* since the boundary form is
* computed as the contravariant
* image of the normal vector to
* the unit cell.
* </ul>
*/
virtual UpdateFlags update_each (const UpdateFlags flags) const;
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_data (const UpdateFlags,
const Quadrature<dim>& quadrature) const;
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_face_data (const UpdateFlags flags,
const Quadrature<dim-1>& quadrature) const;
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_subface_data (const UpdateFlags flags,
const Quadrature<dim-1>& quadrature) const;
/**
* Computes the support points of
* the mapping. For @p MappingQ1
* these are the
* vertices. However, other
* classes may override this
* function. In particular, the
* MappingQ1Eulerian class does
* exactly this by not computing
* the support points from the
* geometry of the current cell
* but instead evaluating an
* externally given displacement
* field in addition to the
* geometry of the cell.
*/
virtual void compute_mapping_support_points(
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
std::vector<Point<spacedim> > &a) const;
/**
* Number of shape functions. Is
* simply the number of vertices
* per cell for the Q1 mapping.
*/
static const unsigned int n_shape_functions = GeometryInfo<dim>::vertices_per_cell;
};
/**
* In order to avoid creation of static MappingQ1 objects at several
* places in the library (in particular in backward compatibility
* functions), we define a static MappingQ1 objects once and for all
* places where it is needed.
*/
template <int dim, int spacedim=dim>
struct StaticMappingQ1
{
static MappingQ1<dim, spacedim> mapping;
};
/*@}*/
/*----------------------------------------------------------------------*/
#ifndef DOXYGEN
template<int dim, int spacedim>
inline
double
MappingQ1<dim,spacedim>::InternalData::shape (const unsigned int qpoint,
const unsigned int shape_nr) const
{
Assert(qpoint*n_shape_functions + shape_nr < shape_values.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_values.size()));
return shape_values [qpoint*n_shape_functions + shape_nr];
}
template<int dim, int spacedim>
inline
double &
MappingQ1<dim,spacedim>::InternalData::shape (const unsigned int qpoint,
const unsigned int shape_nr)
{
Assert(qpoint*n_shape_functions + shape_nr < shape_values.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_values.size()));
return shape_values [qpoint*n_shape_functions + shape_nr];
}
template<int dim, int spacedim>
inline
Tensor<1,dim>
MappingQ1<dim,spacedim>::InternalData::derivative (const unsigned int qpoint,
const unsigned int shape_nr) const
{
Assert(qpoint*n_shape_functions + shape_nr < shape_derivatives.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_derivatives.size()));
return shape_derivatives [qpoint*n_shape_functions + shape_nr];
}
template<int dim, int spacedim>
inline
Tensor<1,dim> &
MappingQ1<dim,spacedim>::InternalData::derivative (const unsigned int qpoint,
const unsigned int shape_nr)
{
Assert(qpoint*n_shape_functions + shape_nr < shape_derivatives.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_derivatives.size()));
return shape_derivatives [qpoint*n_shape_functions + shape_nr];
}
template <int dim, int spacedim>
inline
Tensor<2,dim>
MappingQ1<dim,spacedim>::InternalData::second_derivative (const unsigned int qpoint,
const unsigned int shape_nr) const
{
Assert(qpoint*n_shape_functions + shape_nr < shape_second_derivatives.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_second_derivatives.size()));
return shape_second_derivatives [qpoint*n_shape_functions + shape_nr];
}
template <int dim, int spacedim>
inline
Tensor<2,dim> &
MappingQ1<dim,spacedim>::InternalData::second_derivative (const unsigned int qpoint,
const unsigned int shape_nr)
{
Assert(qpoint*n_shape_functions + shape_nr < shape_second_derivatives.size(),
ExcIndexRange(qpoint*n_shape_functions + shape_nr, 0,
shape_second_derivatives.size()));
return shape_second_derivatives [qpoint*n_shape_functions + shape_nr];
}
template <int dim, int spacedim>
inline
bool
MappingQ1<dim,spacedim>::preserves_vertex_locations () const
{
return true;
}
#endif // DOXYGEN
/* -------------- declaration of explicit specializations ------------- */
DEAL_II_NAMESPACE_CLOSE
#endif
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